A302106 - OEIS

%I #16 Apr 05 2018 18:58:29

%S 1,1,3,45,1095,35151,1381725,64175631,3451040205,211905683649,

%T 14692740175359,1138932255875229,97794610588861299,

%U 9222742814262416391,947982726689249684721,105483448180468629244791,12630764358798125298488577,1619156615552164362662455257,221209937165123652082161577203

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

%H Paul D. Hanna, <a href="/A302106/b302106.txt">Table of n, a(n) for n = 0..100</a>

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

%F G.f.: A(x) = 1 + x/2 * Sum_{n>=0} A(x)^n / 2^n * Product_{k=1..n} (2 - A(x)^(4*k-3)) / (2 - A(x)^(4*k-1)), due to a q-series identity.

%e G.f.: A(x) = 1 + x + 3*x^2 + 45*x^3 + 1095*x^4 + 35151*x^5 + 1381725*x^6 + 64175631*x^7 + 3451040205*x^8 + 211905683649*x^9 + ...

%e such that

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

%o (PARI) /* Continued fraction expression: */

%o {a(n) = my(CF=1, A); for(i=0,n, A = 1 + x*CF +x*O(x^n); for(k=0, n, CF = 1/(2 - A^(4*n-4*k+1)/(1 - A^(2*n-2*k+1)*(A^(2*n-2*k+2) - 1)*CF)) )); polcoeff(A, n)}

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 05 2018