A338693 -id:A338693

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

A360712

Expansion of Sum_{k>0} (k * x * (1 + k*x^k))^k.

+10
5

1, 5, 27, 272, 3125, 46915, 823543, 16781312, 387421218, 10000078125, 285311670611, 8916102153177, 302875106592253, 11112006865911623, 437893890381640625, 18446744074783358976, 827240261886336764177, 39346408075327943829273

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

OFFSET

1,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..386

FORMULA

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

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

MATHEMATICA

a[n_] := DivisorSum[n, #^(#+n/#-1) * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 09 2023 *)

PROG

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

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

CROSSREFS

Cf. A217668, A327249, A338693, A359700, A360618.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Feb 17 2023

STATUS

approved

A360759

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

+10
1

1, 16, 243, 4112, 78125, 1680345, 40353607, 1073766400, 31381060338, 1000000781250, 34522712143931, 1283918489808640, 51185893014090757, 2177953338656796883, 98526125335697265625, 4722366482899710050304, 239072435685151324847153

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

OFFSET

1,2

LINKS

Winston de Greef, Table of n, a(n) for n = 1..383

FORMULA

G.f.: Sum_{k>0} k^k * ( (1 + k*x^k)^k - 1 ).

If p is prime, a(p) = p^(p+2).

MATHEMATICA

a[n_] := DivisorSum[n, #^(# + n/#) * Binomial[#, n/#] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)

PROG

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

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

CROSSREFS

Cf. A318636, A327238, A338685, A338693, A338694.

Cf. A360712.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Feb 19 2023

STATUS

approved

A360770

Expansion of Sum_{k>0} (x * (k + x^k))^k.

+10
0

1, 5, 27, 260, 3125, 46684, 823543, 16777472, 387420498, 10000003125, 285311670611, 8916100495009, 302875106592253, 11112006826381559, 437893890380860625, 18446744073726328848, 827240261886336764177, 39346408075296925015353

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

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..18.

FORMULA

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

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

MATHEMATICA

a[n_] := DivisorSum[n, #^(# - n/# + 1) * Binomial[#, n/# - 1] &]; Array[a, 20] (* Amiram Eldar, Aug 02 2023 *)

PROG

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

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

CROSSREFS

Cf. A260148, A338693, A360712, A360771.

Cf. A360774, A360775, A360776.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Feb 20 2023

STATUS

approved