cp's OEIS Frontend

A193112 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(2*n+1).

%I A193112 #8 Oct 04 2024 16:50:59
%S A193112 1,1,3,13,63,328,1796,10200,59529,354837,2151079,13221261,82200739,
%T A193112 516053099,3266812048,20829635112,133651716406,862342656359,
%U A193112 5591505085491,36416212224801,238114435569354,1562560513492974,10287406857203911
%N A193112 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(2*n+1).
%F A193112 G.f. A(x) satisfies the continued fraction:
%F A193112 1 = A(x)/(1+ x*A(x)^2/(1- x*(1+x)*A(x)^2/(1+ x^3*A(x)^2/(1+ x^2*(1-x^2)*A(x)^2/(1+ x^5*A(x)^2/(1- x^3*(1+x^3)*A(x)^2/(1+ x^7*A(x)^2/(1+ x^4*(1-x^4)*A(x)^2/(1- ...)))))))))
%F A193112 due to an identity of a partial elliptic theta function.
%e A193112 G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 63*x^4 + 328*x^5 + 1796*x^6 +...
%e A193112 which satisfies:
%e A193112 1 = A(x) - x*A(x)^3 - x^3*A(x)^5 + x^6*A(x)^7 + x^10*A(x)^9 - x^15*A(x)^11 - x^21*A(x)^13 ++--...
%e A193112 Related expansions.
%e A193112 A(x)^3 = 1 + 3*x + 12*x^2 + 58*x^3 + 303*x^4 + 1662*x^5 + 9447*x^6 +...
%e A193112 A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 760*x^4 + 4401*x^5 +...
%o A193112 (PARI) {a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=polcoeff(1-sum(m=0, sqrtint(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(2*m+1)), #A-1)); if(n<0, 0, A[n+1])}
%Y A193112 Cf. A193111, A193113, A193114, A193115, A193116.
%K A193112 nonn
%O A193112 0,3
%A A193112 _Paul D. Hanna_, Jul 16 2011