The On-Line Encyclopedia of Integer Sequences (OEIS)

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A171803

G.f. satisfies: A(x) = P(x*A(x))^2 where A(x/P(x)^2) = P(x)^2 and P(x) is the g.f. for Partition numbers (A000041).
(history; published version)

#10 by Vaclav Kotesovec at Tue Oct 03 05:28:53 EDT 2023

STATUS

editing

approved

#9 by Vaclav Kotesovec at Tue Oct 03 05:28:35 EDT 2023

MATHEMATICA

nmax = 30; A[_] = 0; Do[A[x_] = x/Product[(1 - A[x]^k)^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x]/x, x] (* Vaclav Kotesovec, Oct 03 2023 *)

#8 by Vaclav Kotesovec at Tue Oct 03 05:27:02 EDT 2023

MATHEMATICA

(* Calculation of constants {d, c}: *) eq = FindRoot[{r/QPochhammer[s]^2 == s, 1/s + 2*Sqrt[s/r]*Derivative[0, 1][QPochhammer][s, s] == (2*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/8}, {s, 1/4}, WorkingPrecision -> 1200]; {N[1/r /. eq, 120], val = -s*Log[s]*Sqrt[(-1 + s)/(Pi*r*(r*(-8*s*Log[-1 + 1/s] + 4*(-1 + s)*Log[1 - s]^2 + 3*(-1 + s)*Log[s]^2 + 8*Log[1 - s]*(1 + Log[s] - s*Log[s])) + 8*r*(-1 + s)*(-1 + Log[-1 + 1/s])* QPolyGamma[0, 1, s] + 4*r*(-1 + s)*QPolyGamma[0, 1, s]^2 - 4*r*(-1 + s)*QPolyGamma[1, 1, s] - 4*Sqrt[r]*(-1 + s)*s^(5/2)*Log[s]^2* Derivative[0, 2][QPochhammer][s, s] + 8*r*(-1 + s)*s*Log[s]* Derivative[0, 0, 1][QPolyGamma][0, 1, s]))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

STATUS

approved

editing

#7 by Vaclav Kotesovec at Sun May 13 06:58:49 EDT 2018

STATUS

editing

approved

#6 by Vaclav Kotesovec at Sun May 13 06:58:46 EDT 2018

CROSSREFS

Cf. A171802, A171804, A171805, A109085, A000041, A304444.

STATUS

approved

editing

#5 by Vaclav Kotesovec at Sat Nov 11 10:29:40 EST 2017

STATUS

editing

approved

#4 by Vaclav Kotesovec at Sat Nov 11 10:29:34 EST 2017

FORMULA

From Vaclav Kotesovec, Nov 11 2017: (Start)

a(n) ~ c * d^n / n^(3/2), where

d = 8.4251672106325154177760155558415141093613298032469849432733825... and

c = 0.6057593757525562292332998445991464666128350560350232598293... (End)

#3 by Vaclav Kotesovec at Sat Nov 11 10:16:19 EST 2017

MATHEMATICA

nmax = 25; Rest[CoefficientList[InverseSeries[Series[x*Product[(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}]], x]] (* Vaclav Kotesovec, Nov 11 2017 *)

STATUS

approved

editing

#2 by Russ Cox at Fri Mar 30 18:37:20 EDT 2012

AUTHOR

_Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 19 2009

Discussion

Fri Mar 30

18:37

OEIS Server: https://oeis.org/edit/global/213

#1 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010

NAME

G.f. satisfies: A(x) = P(x*A(x))^2 where A(x/P(x)^2) = P(x)^2 and P(x) is the g.f. for Partition numbers (A000041).

DATA

1, 2, 9, 48, 286, 1818, 12086, 82992, 584079, 4190738, 30539814, 225426240, 1681904909, 12663614266, 96099303213, 734250983952, 5643749482600, 43610375803722, 338578974873523, 2639771240159904, 20659895819582337

OFFSET

0,2

FORMULA

G.f. satisfies: A(x) = 1/Product_{n>=1} (1 - A(x)^n)^2.

G.f.: A(x) = Series_Reversion(x*eta(x)^2) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).

Self-convolution of A171802.

EXAMPLE

G.f.: A(x) = 1 + 2*x + 9*x^2 + 48*x^3 + 286*x^4 + 1818*x^5 +...

A(x/P(x)^2) = P(x)^2 where:

P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...

P(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 +...

PROG

(PARI) a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))^2), n)

CROSSREFS

Cf. A171802, A171804, A171805, A109085, A000041.

KEYWORD

nonn

AUTHOR

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 19 2009

STATUS

approved