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A301929 G.f. A(x) satisfies: x = Sum_{n>=1} (1+x)^(n^2) * x^n / A(x)^n. 4

%I #13 May 07 2018 02:16:44

%S 1,2,3,5,12,37,138,595,2843,14844,83212,496473,3128584,20707672,

%T 143342216,1034075244,7752274237,60251286521,484483164365,

%U 4023459643530,34455215830001,303839675537827,2755675307738286,25675275100067189,245502965520844801,2406797239543382867,24170220195274548727,248441483165679473094,2611787614440970964621

%N G.f. A(x) satisfies: x = Sum_{n>=1} (1+x)^(n^2) * x^n / A(x)^n.

%H Paul D. Hanna, <a href="/A301929/b301929.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: x = Sum_{n>=1} x^n/A(x)^n * (1+x)^n * Product_{k=1..n} (A(x) - x*(1+x)^(4*k-3)) / (A(x) - x*(1+x)^(4*k-1)), due to a q-series identity.

%F G.f.: 1+x = 1/(1 - q*x/(A(x) - q*(q^2-1)*x/(1 - q^5*x/(A(x) - q^3*(q^4-1)*x/(1 - q^9*x/(A(x) - q^5*(q^6-1)*x/(1 - q^13*x/(A(x) - q^7*(q^8-1)*x/(1 - ...))))))))), where q = (1+x), a continued fraction due to a partial elliptic theta function identity.

%e G.f.: A(x) = 1 + 2*x + 3*x^2 + 5*x^3 + 12*x^4 + 37*x^5 + 138*x^6 + 595*x^7 + 2843*x^8 + 14844*x^9 + 83212*x^10 + 496473*x^11 + 3128584*x^12 + ...

%e such that

%e x = (1+x)*x/A(x) + (1+x)^4*x^2/A(x)^2 + (1+x)^9*x^3/A(x)^3 + (1+x)^16*x^4/A(x)^4 + (1+x)^25*x^5/A(x)^5 + (1+x)^36*x^6/A(x)^6 + (1+x)^49*x^7/A(x)^7 + ...

%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, ((1+x)^n +x*O(x^#A))^n * x^n/Ser(A)^n ) )[#A+1] ); A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A301927, A303058, A107595.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 04 2018

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