A193111 - OEIS

%I #10 Oct 23 2020 03:59:40

%S 1,1,2,6,19,63,218,781,2869,10742,40846,157318,612446,2406100,9527159,

%T 37981611,152328497,614167702,2487941464,10121128882,41330709103,

%U 169362297620,696187639438,2870017515884,11862845007114,49152859179055

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

%H Paul D. Hanna, <a href="/A193111/b193111.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies the continued fraction:

%F 1 = A(x)/(1+ x*A(x)/(1- x*(1+x)*A(x)/(1+ x^3*A(x)/(1+ x^2*(1-x^2)*A(x)/(1+ x^5*A(x)/(1- x^3*(1+x^3)*A(x)/(1+ x^7*A(x)/(1+ x^4*(1-x^4)*A(x)/(1- ...)))))))))

%F due to an identity of a partial elliptic theta function.

%F a(n) ~ c * d^n / n^(3/2), where d = 4.39601711776597002671715735353... and c = 0.541742533522963093430641871... - _Vaclav Kotesovec_, Oct 23 2020

%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 63*x^5 + 218*x^6 +...

%e which satisfies:

%e 1 = A(x) - x*A(x)^2 - x^3*A(x)^3 + x^6*A(x)^4 + x^10*A(x)^5 - x^15*A(x)^6 - x^21*A(x)^7 ++--...

%e Related expansions.

%e A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 54*x^4 + 188*x^5 + 674*x^6 +...

%e A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 111*x^4 + 405*x^5 + 1505*x^6 +...

%o (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)^(m+1)), #A-1)); if(n<0, 0, A[n+1])}

%Y Cf. A193112, A193113, A193114, A193115, A193116.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 16 2011