A005758 Number of partitions of n into parts of 12 kinds.

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

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy
• nice

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

Examples

			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 + ...
		

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

Crossrefs

12th column of A144064.

Cf. A006922, A186209.

Programs

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 *)

PARI

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

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