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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). 4
1, 2, 9, 48, 286, 1818, 12086, 82992, 584079, 4190738, 30539814, 225426240, 1681904909, 12663614266, 96099303213, 734250983952, 5643749482600, 43610375803722, 338578974873523, 2639771240159904, 20659895819582337 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

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.

From Vaclav Kotesovec, Nov 11 2017: (Start)

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

d = 8.4251672106325154177760155558415141093613298032469849432733825... and

c = 0.6057593757525562292332998445991464666128350560350232598293... (End)

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 +...

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 *)

PROG

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

CROSSREFS

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

Sequence in context: A153297 A153390 A118341 * A100427 A214404 A074143

Adjacent sequences:  A171800 A171801 A171802 * A171804 A171805 A171806

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 19 2009

STATUS

approved

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Last modified July 11 08:10 EDT 2020. Contains 335626 sequences. (Running on oeis4.)