A193111 G.f. satisfies: 1 = Sum_{n>=0} (-x)^(n*(n+1)/2) * A(x)^(n+1).
1, 1, 2, 6, 19, 63, 218, 781, 2869, 10742, 40846, 157318, 612446, 2406100, 9527159, 37981611, 152328497, 614167702, 2487941464, 10121128882, 41330709103, 169362297620, 696187639438, 2870017515884, 11862845007114, 49152859179055
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 63*x^5 + 218*x^6 +... which satisfies: 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 ++--... Related expansions. A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 54*x^4 + 188*x^5 + 674*x^6 +... A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 111*x^4 + 405*x^5 + 1505*x^6 +...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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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(2*(#A))+1, (-x)^(m*(m+1)/2)*Ser(A)^(m+1)), #A-1)); if(n<0, 0, A[n+1])}
Formula
G.f. A(x) satisfies the continued fraction:
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- ...)))))))))
due to an identity of a partial elliptic theta function.
a(n) ~ c * d^n / n^(3/2), where d = 4.39601711776597002671715735353... and c = 0.541742533522963093430641871... - Vaclav Kotesovec, Oct 23 2020