A005758 Number of partitions of n into parts of 12 kinds. (Formerly M4854)

1, 12, 90, 520, 2535, 10908, 42614, 153960, 521235, 1669720, 5098938, 14931072, 42124380, 114945780, 304351020, 784087848, 1970043621, 4837060800, 11626305640, 27398234760, 63388751544, 144156086776, 322590526350

Offset

0,2

Comments

Euler transform of A010851. - Alois P. Heinz, Oct 17 2008

Convolution square of A005758 = A006922: (1, 24, 324, 3200, 25650, ...). - Gary W. Adamson, Jun 13 2009

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)

Roland Bacher, P. De La Harpe, Conjugacy growth series of some infinitely generated groups. 2016, hal-01285685v2.

N. J. A. Sloane, Transforms

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

Formula

G.f.: Product ( 1 - x^k )^(-12).

Expansion of q^(1/2) * eta(q)^-12 in powers of q. - Michael Somos, Mar 07 2012

Convolution inverse of A000735.

a(n) ~ exp(2 * Pi * sqrt(2*n)) / (2^(15/4) * n^(15/4)). - Vaclav Kotesovec, Feb 28 2015

a(0) = 1, a(n) = (12/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017

G.f.: exp(12*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

Example

G.f. = 1 + 12*x + 90*x^2 + 520*x^3 + 2535*x^4 + 10908*x^5 + 42614*x^6 + ...
G.f. = 1/q + 12*q + 90*q^3 + 520*q^5 + 2535*q^7 + 10908*q^9 + 42614*q^11 + ...

Maple

with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*12, 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, x]^12, {x, 0, 30}], x]  (* Harvey P. Dale, Apr 21 2011 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x * O(x^n))^12, n))}; /* Michael Somos, Mar 07 2012 */

Crossrefs

12th column of A144064. - Alois P. Heinz, Oct 17 2008

Cf. A006922. - Gary W. Adamson, Jun 13 2009

Sequence in context: A121590 A341388 A186209 * A084485 A130072 A135158

Adjacent sequences: A005755 A005756 A005757 * A005759 A005760 A005761

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Simon Plouffe

Status

approved