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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

%I

%S 1,2,9,48,286,1818,12086,82992,584079,4190738,30539814,225426240,

%T 1681904909,12663614266,96099303213,734250983952,5643749482600,

%U 43610375803722,338578974873523,2639771240159904,20659895819582337

%N 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).

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

%F 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).

%F Self-convolution of A171802.

%F From _Vaclav Kotesovec_, Nov 11 2017: (Start)

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

%F d = 8.4251672106325154177760155558415141093613298032469849432733825... and

%F c = 0.6057593757525562292332998445991464666128350560350232598293... (End)

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

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

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

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

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

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

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 19 2009

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Last modified August 14 07:04 EDT 2020. Contains 336477 sequences. (Running on oeis4.)