A010857 -id:A010857

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A023016

Number of partitions of n into parts of 18 kinds.

+10
2

1, 18, 189, 1482, 9576, 53676, 269325, 1235286, 5256711, 20985272, 79260723, 285139764, 982349361, 3255488082, 10416507579, 32281134120, 97154549289, 284625019800, 813310723925, 2270826800172, 6204926551824, 16615751700618

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

OFFSET

0,2

COMMENTS

a(n) is Euler transform of A010857. - Alois P. Heinz, Oct 17 2008

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

FORMULA

a(0) = 1, a(n) = (18/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017

MAPLE

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*18, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008

MATHEMATICA

CoefficientList[Series[1/QPochhammer[x]^18, {x, 0, 30}], x] (* Indranil Ghosh, Mar 27 2017 *)

PROG

(PARI) Vec(1/eta(x)^18 + O(x^30)) \\ Indranil Ghosh, Mar 27 2017

CROSSREFS

Cf. 18th column of A144064. - Alois P. Heinz, Oct 17 2008

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

A249693

a(4n) = 3*n+1, a(2n+1) = 3*n+2, a(4n+2) = 3*n.

+10
1

1, 2, 0, 5, 4, 8, 3, 11, 7, 14, 6, 17, 10, 20, 9, 23, 13, 26, 12, 29, 16, 32, 15, 35, 19, 38, 18, 41, 22, 44, 21, 47, 25, 50, 24, 53, 28, 56, 27, 59, 31, 62, 30, 65, 34, 68, 33, 71, 37, 74, 36, 77, 40, 80, 39, 83, 43, 86, 42, 89, 46, 92, 45

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

OFFSET

0,2

COMMENTS

A permutation of the nonnegative numbers.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

FORMULA

a(n+4) = a(n) + (sequence of period 2: repeat 3, 6).

a(4n+1) = 2*a(4n).

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).

a(n) is the rank of A061037(n) = -1, -3, 0, 5, ... in A247829(n) = 0, -1, -3, 2, ... .

G.f.: (1 + 2*x - x^2 + 3*x^3 + 3*x^4 + x^5)/(1 - x^2 - x^4 + x^6).

a(n) = (1 + 9*n - 3*(n+1)*(-1)^n + 10*cos(n*Pi/2))/8. - Robert Israel, Dec 03 2014

MATHEMATICA

a[n_] := (1/8)*(3*(-1)^(n+1)*(n+1)+9*n+10*{1, 0, -1, 0}[[Mod[n, 4]+1]]+1); Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Dec 04 2014, after Robert Israel *)

PROG

(PARI) x='x+O('x^75); Vec((1 + 2*x - x^2 + 3*x^3 + 3*x^4 + x^5)/(1 - x^2 - x^4 + x^6)) \\ G. C. Greubel, Sep 20 2018

(Magma) m:=75; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + 2*x - x^2 + 3*x^3 + 3*x^4 + x^5)/(1 - x^2 - x^4 + x^6))); // G. C. Greubel, Sep 20 2018