A355871 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * A(x)^n.

1, 2, 2, 12, 22, 144, 318, 2102, 5120, 34274, 88352, 597002, 1599676, 10879502, 29983958, 204851678, 576914820, 3953960052, 11329537402, 77815428652, 226170428918, 1555598157856, 4576144621100, 31500863667990, 93634976287220, 644808182456240, 1934219875423410

Offset: 0

Author

Paul D. Hanna, Jul 19 2022

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 2*x^2 + 12*x^3 + 22*x^4 + 144*x^5 + 318*x^6 + 2102*x^7 + 5120*x^8 + 34274*x^9 + 88352*x^10 + 597002*x^11 + 1599676*x^12 + ...
where
2 = ... + x^6/A(x)^4 + x^3/A(x)^3 + x/A(x)^2 + 1/A(x) + 1 + x*A(x) + x^3*A(x)^2 + x^6*A(x)^3 + x^10*A(x)^4 + ... + x^(n*(n+1)/2) * A(x)^n + ...
Also,
2 = (1 + x*A(x))*(1 + 1/A(x))*(1-x) * (1 + x^2*A(x))*(1 + x/A(x))*(1-x^2) * (1 + x^3*A(x))*(1 + x^2/A(x))*(1-x^3) * (1 + x^4*A(x))*(1 + x^3/A(x))*(1-x^4) * ...
Specific values.
A(1/5) = 1.8349253975...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A355870.

Programs

PARI

{a(n) = my(A=[1],M); for(i=1,n, A=concat(A,0); M = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( -2 + sum(m=-M,M, x^(m*(m+1)/2) * Ser(A)^m ), #A-1));A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x,y) = Sum_{n>=0} a(n)*x^n satisfies:

(1) 2 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * A(x)^n.

(2) 2 = Product_{n>=1} (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)) * (1 - x^n), by the Jacobi triple product identity.

(3) a(n) = Sum_{k=0..3*n} A355870(n,k)*2^k for n >= 0.