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A026007 Expansion of Product_{m>=1} (1 + q^m)^m; number of partitions of n into distinct parts, where n different parts of size n are available. 32
1, 1, 2, 5, 8, 16, 28, 49, 83, 142, 235, 385, 627, 1004, 1599, 2521, 3940, 6111, 9421, 14409, 21916, 33134, 49808, 74484, 110837, 164132, 241960, 355169, 519158, 755894, 1096411, 1584519, 2281926, 3275276, 4685731, 6682699, 9501979, 13471239, 19044780, 26850921, 37756561, 52955699 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equals A000219: (1, 1, 3, 6, 13, 24, 48, 86, ...) convolved with the aerated version of the latter: (1, 0, 1, 0, 3, 0, 6, 0, 13, ...). - Gary W. Adamson, Jun 13 2009

In general, for t > 0, if g.f. = Product_{m>=1} (1 + t*q^m)^m then a(n) ~ c^(1/6) * exp(3^(2/3) * c^(1/3) * n^(2/3) / 2) / (3^(2/3) * (t+1)^(1/12) * sqrt(2*Pi) * n^(2/3)), where c = Pi^2*log(t) + log(t)^3 - 6*polylog(3, -1/t). - Vaclav Kotesovec, Jan 04 2016

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Vaclav Kotesovec, Graph - The asymptotic ratio

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 18.

FORMULA

a(n) = 1/n*Sum_{k=1..n} A078306(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002

G.f. Product_{m>=1} (1+x^m)^m. Weighout transform of natural numbers (A000027). Euler transform of A026741. - Franklin T. Adams-Watters, Mar 16 2006

a(n) ~ Zeta(3)^(1/6) * exp((3/2)^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (2^(3/4) * 3^(1/3) * sqrt(Pi) * n^(2/3)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 05 2015

EXAMPLE

For n = 4, we have 8 partitions

01: [4]

02: [4']

03: [4'']

04: [4''']

05: [3, 1]

06: [3', 1]

07: [3'', 1]

08: [2, 2']

MAPLE

with(numtheory):

b:= proc(n) option remember;

      add((-1)^(n/d+1)*d^2, d=divisors(n))

    end:

a:= proc(n) option remember;

      `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)

    end:

seq(a(n), n=0..45);  # Alois P. Heinz, Aug 03 2013

MATHEMATICA

a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^2, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 41}] (* Jean-Fran├žois Alcover, Apr 17 2014, after Vladeta Jovovic *)

nmax=50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*x^k/(k*(1-x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 28 2015 *)

PROG

(PARI)

N=66; q='q+O('q^N);

gf= prod(n=1, N, (1+q^n)^n );

Vec(gf)

/* Joerg Arndt, Oct 06 2012 */

CROSSREFS

Cf. A000009, A000027, A026741, A073592, A255528, A261562, A266857.

Cf. A000219. - Gary W. Adamson, Jun 13 2009

Cf. A027998, A248882, A248883, A248884.

Cf. A026011, A027346, A027906.

Sequence in context: A137685 A169826 A093065 * A032233 A026530 A032254

Adjacent sequences:  A026004 A026005 A026006 * A026008 A026009 A026010

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 11 09:57 EST 2016. Contains 268164 sequences.