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URL: https://oeis.org/A005758

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A005758
Number of partitions of n into parts of 12 kinds.
8
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
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
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 16 * exp(-Pi/2) * sqrt(2) * Gamma(3/4)^12 / Pi^3 = A388163. - Simon Plouffe, Sep 15 2025
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.
Sequence in context: A121590 A341388 A186209 * A084485 A130072 A135158
KEYWORD
nonn,easy,nice
STATUS
approved