A318636 -id:A318636

A327238

Expansion of Sum_{k>=1} ((1 + k * x^k)^k - 1).

+10
7

1, 4, 9, 20, 25, 63, 49, 160, 108, 350, 121, 940, 169, 1225, 1475, 2304, 289, 7560, 361, 8025, 12446, 7139, 529, 58192, 3750, 13858, 61965, 102655, 841, 191181, 961, 318464, 220704, 40460, 354172, 1304370, 1369, 63175, 629863, 4012608, 1681, 1916733, 1849

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = Sum_{d|n} (n/d)^d * binomial(n/d,d).

a(p) = p^2, where p is prime.

MATHEMATICA

nmax = 43; CoefficientList[Series[Sum[((1 + k x^k)^k - 1), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Table[DivisorSum[n, (n/#)^# Binomial[n/#, #] &], {n, 1, 43}]

PROG

(PARI) a(n)={sumdiv(n, d, (n/d)^d * binomial(n/d, d))} \\ Andrew Howroyd, Sep 14 2019

CROSSREFS

Cf. A055225, A056045, A318636, A327124.

KEYWORD

nonn,look

AUTHOR

Ilya Gutkovskiy, Sep 14 2019

STATUS

approved

A338693

a(n) = Sum_{d|n} d^(d - n/d) * binomial(d, n/d).

+10
7

1, 4, 27, 257, 3125, 46665, 823543, 16777312, 387420490, 10000001250, 285311670611, 8916100467712, 302875106592253, 11112006825910963, 437893890380859625, 18446744073716891649, 827240261886336764177, 39346408075296709766628, 1978419655660313589123979

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Winston de Greef, Table of n, a(n) for n = 1..385

FORMULA

G.f.: Sum_{k>=1} ( (k + x^k)^k - k^k ).

If p is prime, a(p) = p^p.

MATHEMATICA

a[n_] := DivisorSum[n, #^(# - n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)

PROG

(PARI) a(n) = sumdiv(n, d, d^(d-n/d)* binomial(d, n/d));

(PARI) N=20; x='x+O('x^N); Vec(sum(k=1, N, (k+x^k)^k-k^k))

CROSSREFS

Cf. A318636, A318637, A318638, A338685, A338694.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 24 2021

STATUS

approved

A318637

Expansion of Sum_{n>=1} ( (2 + x^n)^n - 2^n ).

+10
6

1, 4, 12, 33, 80, 198, 448, 1048, 2305, 5200, 11264, 24824, 53248, 115360, 245800, 526081, 1114112, 2364064, 4980736, 10497290, 22020656, 46165504, 96468992, 201396028, 419430401, 872574976, 1811944704, 3758469400, 7784628224, 16107002892, 33285996544, 68721443936, 141733963008, 292062232576, 601295421524, 1236960724929, 2542620639232, 5222702645248, 10720238663680, 21990282376768

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1024

FORMULA

a(n) ~ n * 2^(n-1). - Vaclav Kotesovec, Oct 10 2020

a(n) = Sum_{d|n} 2^(d - n/d) * binomial(d, n/d). - Seiichi Manyama, Apr 24 2021

G.f.: Sum_{k >=1} x^(k^2)/(1-2*x^k)^(k+1). - Seiichi Manyama, Oct 30 2023

EXAMPLE

G.f.: A(x) = x + 4*x^2 + 12*x^3 + 33*x^4 + 80*x^5 + 198*x^6 + 448*x^7 + 1048*x^8 + 2305*x^9 + 5200*x^10 + 11264*x^11 + 24824*x^12 + 53248*x^13 + 115360*x^14 + ...

such that

A(x) = x + (2 + x^2)^2 - 2^2 + (2 + x^3)^3 - 2^3 + (2 + x^4)^4 - 2^4 + (2 + x^5)^5 - 2^5 + (2 + x^6)^6 - 2^6 + (2 + x^7)^7 - 2^7 + ...

RELATED SERIES.

The g.f. A(x) equals following series at y = 2:

Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...

PROG

(PARI) {a(n) = polcoeff( sum(m=1, n, (x^m + 2 +x*O(x^n))^m - 2^m), n)}

for(n=1, 100, print1(a(n), ", "))

(PARI) a(n) = sumdiv(n, d, 2^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021

CROSSREFS

Cf. A318636, A318638.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 07 2018

STATUS

approved

A338682

a(n) = Sum_{d|n} (-1)^(d-1) * binomial(d+n/d-1, d).

+10
6

1, 1, 4, 0, 6, 3, 8, -8, 20, 0, 12, -12, 14, -7, 72, -65, 18, 10, 20, -61, 142, -33, 24, -203, 152, -52, 248, -183, 30, 121, 32, -617, 398, -102, 828, -619, 38, -133, 600, -896, 42, 140, 44, -870, 2864, -207, 48, -4438, 1766, 751, 1192, -1587, 54, -348, 4424, -3011, 1598, -348, 60

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

FORMULA

G.f.: Sum_{k >= 1} (1 - 1/(1 + x^k)^k).

G.f.: - Sum_{k >= 1} (-x)^k/(1 - x^k)^(k+1).

If p is prime, a(p) = (-1)^(p-1) + p.

MATHEMATICA

a[n_] := DivisorSum[n, (-1)^(# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 60] (* Amiram Eldar, Apr 24 2021 *)

PROG

(PARI) a(n) = sumdiv(n, d, (-1)^(d-1)*binomial(d+n/d-1, d));

(PARI) N=66; x='x+O('x^N); Vec(sum(k=1, N, 1-1/(1+x^k)^k))

(PARI) N=66; x='x+O('x^N); Vec(-sum(k=1, N, (-x)^k/(1-x^k)^(k+1)))

CROSSREFS

Cf. A081543, A217670, A318636, A338683, A338684.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Apr 23 2021

STATUS

approved

A318638

Expansion of Sum_{n>=1} ( (3 + x^n)^n - 3^n ).

+10
5

1, 6, 27, 109, 405, 1467, 5103, 17550, 59050, 197100, 649539, 2126991, 6908733, 22325625, 71744625, 229602925, 731794257, 2324602206, 7360989291, 23245524600, 73222475256, 230128853031, 721764371007, 2259440202825, 7060738412026, 22029517662984, 68630377426119, 213516777941712, 663426981193869, 2058911488612863, 6382625094934119, 19765549255048254, 61149666233193318

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1024

FORMULA

a(n) ~ n * 3^(n-1). - Vaclav Kotesovec, Oct 10 2020

a(n) = Sum_{d|n} 3^(d - n/d) * binomial(d, n/d). - Seiichi Manyama, Apr 24 2021

G.f.: Sum_{k >=1} x^(k^2)/(1-3*x^k)^(k+1). - Seiichi Manyama, Oct 30 2023

EXAMPLE

G.f.: A(x) = x + 6*x^2 + 27*x^3 + 109*x^4 + 405*x^5 + 1467*x^6 + 5103*x^7 + 17550*x^8 + 59050*x^9 + 197100*x^10 + 649539*x^11 + 2126991*x^12 + ...

such that

A(x) = x + (3 + x^2)^2 - 3^2 + (3 + x^3)^3 - 3^3 + (3 + x^4)^4 - 3^4 + (3 + x^5)^5 - 3^5 + (3 + x^6)^6 - 3^6 + (3 + x^7)^7 - 3^7 + ...

RELATED SERIES.

The g.f. A(x) equals following series at y = 3:

Sum_{n>=1} ((y + x^n)^n - y^n) = x + 2*y*x^2 + 3*y^2*x^3 + (4*y^3 + 1)*x^4 + 5*y^4*x^5 + (6*y^5 + 3*y)*x^6 + 7*y^6*x^7 + (8*y^7 + 6*y^2)*x^8 + (9*y^8 + 1)*x^9 + (10*y^9 + 10*y^3)*x^10 + 11*y^10*x^11 + (12*y^11 + 15*y^4 + 4*y)*x^12 + 13*y^12*x^13 + (14*y^13 + 21*y^5)*x^14 + (15*y^14 + 10*y^2)*x^15 + (16*y^15 + 28*y^6 + 1)*x^16 + ...

PROG

(PARI) {a(n) = polcoeff( sum(m=1, n, (x^m + 3 +x*O(x^n))^m - 3^m), n)}

for(n=1, 100, print1(a(n), ", "))

(PARI) a(n) = sumdiv(n, d, 3^(d-n/d)* binomial(d, n/d)); \\ Seiichi Manyama, Apr 24 2021

CROSSREFS

Cf. A318636, A318637, A338693.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 07 2018

STATUS

approved

A338685

a(n) = Sum_{d|n} d^n * binomial(d, n/d).

+10
5

1, 8, 81, 1040, 15625, 282123, 5764801, 134610944, 3486804084, 100097656250, 3138428376721, 107025924222976, 3937376385699289, 155582338242342053, 6568408660888671875, 295155786482995691520, 14063084452067724991009, 708240750793407501694308, 37589973457545958193355601

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Winston de Greef, Table of n, a(n) for n = 1..384

FORMULA

G.f.: Sum_{k >= 1} ((1 + (k * x)^k)^k - 1).

If p is prime, a(p) = p^(p+1).

MATHEMATICA

a[n_] := DivisorSum[n, #^n * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)

PROG

(PARI) a(n) = sumdiv(n, d, d^n*binomial(d, n/d));

(PARI) N=20; x='x+O('x^N); Vec(sum(k=1, N, (1+(k*x)^k)^k-1))

CROSSREFS

Cf. A023887, A318636, A327238, A338684, A338693.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 23 2021

STATUS

approved

A338694

a(n) = Sum_{d|n} d^d * binomial(d, n/d).

+10
5

1, 8, 81, 1028, 15625, 280017, 5764801, 134219264, 3486784428, 100000031250, 3138428376721, 106993206079936, 3937376385699289, 155568095575106627, 6568408355712921875, 295147905179822588160, 14063084452067724991009, 708235345355351624428356, 37589973457545958193355601

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Winston de Greef, Table of n, a(n) for n = 1..384

FORMULA

G.f.: Sum_{k>=1} ( (k + k * x^k)^k - k^k ) = Sum_{k>=1} k^k * ( (1 + x^k)^k - 1 ).

If p is prime, a(p) = p^(p+1).

MATHEMATICA

a[n_] := DivisorSum[n, #^# * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Apr 24 2021 *)

PROG

(PARI) a(n) = sumdiv(n, d, d^d*binomial(d, n/d));

(PARI) N=20; x='x+O('x^N); Vec(sum(k=1, N, (k+k*x^k)^k-k^k))

CROSSREFS

Cf. A062796, A318636, A318637, A318638, A338693.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 24 2021

STATUS

approved

A327124

Expansion of Sum_{k>=1} ((1 - (-x)^k)^k - 1).

+10
3

1, -2, 3, -3, 5, -3, 7, -2, 10, 0, 11, -1, 13, 7, 25, 13, 17, -2, 19, 30, 56, 33, 23, 1, 26, 52, 111, 98, 29, -51, 31, 158, 198, 102, 56, 24, 37, 133, 325, 304, 41, -189, 43, 517, 626, 207, 47, 191, 50, -2, 731, 988, 53, -435, 517, 1315, 1026, 348, 59, 18

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = Sum_{d|n} (-1)^(n-d) * binomial(n/d,d).

a(p) = p, where p is odd prime.

MATHEMATICA

nmax = 60; CoefficientList[Series[Sum[((1 - (-x)^k)^k - 1), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Table[DivisorSum[n, (-1)^(n - #) Binomial[n/#, #] &], {n, 1, 60}]

PROG

(PARI) a(n)={sumdiv(n, d, (-1)^(n-d) * binomial(n/d, d))} \\ Andrew Howroyd, Sep 14 2019

CROSSREFS

Cf. A056045, A112329, A318636, A327238.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Sep 14 2019

STATUS

approved

A340625

a(n) = Sum_{d|n, d odd, d <= n/d} binomial(n/d, d).

+10
2

1, 2, 3, 4, 5, 6, 7, 8, 10, 10, 11, 16, 13, 14, 25, 16, 17, 38, 19, 20, 56, 22, 23, 80, 26, 26, 111, 28, 29, 156, 31, 32, 198, 34, 56, 256, 37, 38, 325, 96, 41, 406, 43, 44, 626, 46, 47, 608, 50, 302, 731, 52, 53, 870, 517, 64, 1026, 58, 59, 1992, 61, 62, 1429, 64, 1352, 1606, 67, 68, 1840, 2192, 71, 2096, 73, 74

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..10000

FORMULA

G.f.: (1/2) * Sum_{k >= 1} ((1 + x^k)^k - (1 - x^k)^k).

If p is prime, a(p) = p.

MATHEMATICA

a[n_] := DivisorSum[n, Binomial[n/#, #] &, OddQ[#] &]; Array[a, 75] (* Amiram Eldar, Apr 25 2021 *)

PROG

(PARI) a(n) = sumdiv(n, d, (d%2)*binomial(n/d, d));

(PARI) N=99; x='x+O('x^N); Vec(sum(k=1, N, (1+x^k)^k-(1-x^k)^k)/2)

CROSSREFS

Cf. A318636, A327124, A340626.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Apr 25 2021

STATUS

approved

A376017

a(n) = Sum_{d|n} d^(n/d - d) * binomial(n/d,d).

+10
2

1, 2, 3, 5, 5, 12, 7, 32, 10, 90, 11, 264, 13, 686, 105, 1809, 17, 5166, 19, 11560, 2856, 28182, 23, 81456, 26, 159770, 61263, 375004, 29, 1122660, 31, 1984032, 1082598, 4456482, 560, 14486329, 37, 22413350, 16888053, 50674560, 41, 174582072, 43, 247627820, 241884450

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..5000

FORMULA

G.f.: Sum_{k>=1} x^(k^2) / (1 - k*x^k)^(k+1).

If p is prime, a(p) = p.

PROG

(PARI) a(n) = sumdiv(n, d, d^(n/d-d)*binomial(n/d, d));

(PARI) my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, x^k^2/(1-k*x^k)^(k+1)))

(Python)

from math import comb

from itertools import takewhile

from sympy import divisors

def A376017(n): return sum(d**((m:=n//d)-d)*comb(m, d) for d in takewhile(lambda d:d**2<=n, divisors(n))) # Chai Wah Wu, Sep 06 2024

CROSSREFS

Cf. A318636.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Sep 06 2024

STATUS

approved