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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. 79
 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 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 Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, arXiv:2303.02240 [math.CO], 2023. 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-François 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, A285223, A304040. Cf. A000219. - Gary W. Adamson, Jun 13 2009 Cf. A027998, A248882, A248883, A248884. Cf. A026011, A027346, A027906. Column k=1 of A284992. Sequence in context: A169826 A093065 A301596 * A032233 A026530 A336135 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 26 12:26 EST 2024. Contains 370352 sequences. (Running on oeis4.)