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 A301927 G.f. A(x) satisfies: x = Sum_{n>=1} x^n / ( (1-x)^(n^2) * A(x)^n ). 1

%I #6 May 06 2018 16:31:52

%S 1,2,4,9,24,77,294,1296,6403,34644,201932,1253513,8219110,56578239,

%T 406990651,3048202700,23700070773,190830842843,1588016365186,

%U 13633603416558,120574656241999,1097006289005674,10255338612462641,98403208150304070,968186766428157206,9759036265967791137,100690787844977985900,1062601625749170026894,11461320511629994319890

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

%F G.f.: x = Sum_{n>=1} x^n/A(x)^n * (1-x)^n * Product_{k=1..n} (x - (1-x)^(4*k-3)*A(x)) / (x - (1-x)^(4*k-1)*A(x)), 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/(1-x), a continued fraction due to a partial elliptic theta function identity.

%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 9*x^3 + 24*x^4 + 77*x^5 + 294*x^6 + 1296*x^7 + 6403*x^8 + 34644*x^9 + 201932*x^10 + 1253513*x^11 + 8219110*x^12 + ...

%e such that

%e x = x/((1-x)*A(x)) + x^2/((1-x)^4*A(x)^2) + x^3/((1-x)^9*A(x)^3) + x^4/((1-x)^16*A(x)^4) + x^5/((1-x)^25*A(x)^5) + x^6/((1-x)^36*A(x)^6) + x^7/((1-x)^49*A(x)^7) + x^8/((1-x)^64*A(x)^8) + ...

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

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

%Y Cf. A301929.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 06 2018

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