A193114 G.f. A(x) satisfies 1 = Sum_{n>=0} (-x)^(n^2) * A(x)^(n+1).
1, 1, 2, 5, 13, 37, 111, 345, 1103, 3604, 11977, 40356, 137543, 473317, 1642258, 5738828, 20179338, 71346433, 253485527, 904536366, 3240418665, 11649734335, 42017535527, 151992797355, 551298507620, 2004602732825, 7305747551718, 26682235709115
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 111*x^6 + ... which satisfies: 1 = A(x) - x*A(x)^2 + x^4*A(x)^3 - x^9*A(x)^4 + x^16*A(x)^5 -+ ... Related expansions. A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 40*x^4 + 120*x^5 + 373*x^6 + ... A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 87*x^4 + 276*x^5 + 893*x^6 + ...
Links
- Robert Israel, Table of n, a(n) for n = 0..1600
Programs
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Maple
e36:= 1 - add((-x)^(n^2)*a^(n+1),n=0..6): S:= series(RootOf(e36,a),x,37): seq(coeff(S,x,i),i=0..36); # Robert Israel, Apr 10 2023
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PARI
{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(#A)+1, (-x)^(m^2)*Ser(A)^(m+1)), #A-1)); if(n<0, 0, A[n+1])}
Formula
The g.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n>=0} (-x)^n*A(x)^(n+1) * Product_{k=1..n} (1 + x^(4*k-3)*A(x))/(1 + x^(4*k-1)*A(x));
(2) 1 = A(x)/(1 + x*A(x)/(1 - x*(1-x^2)*A(x)/(1 + x^5*A(x)/(1 - x^3*(1-x^4)*A(x)/(1 + x^9*A(x)/(1 - x^5*(1-x^6)*A(x)/(1 + x^13*A(x)/(1 - x^7*(1-x^8)*A(x)/(1- ...))))))))) (continued fraction);
due to identities of a partial elliptic theta function.