A355353 G.f. A(x) satisfies: 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 3, 18, 108, 660, 4275, 29106, 205377, 1485279, 10943424, 81866493, 620316297, 4751289063, 36727782675, 286153810542, 2244799306134, 17715992048886, 140560480602810, 1120518766292436, 8970573523101477, 72091628161825608, 581375787259765554, 4703286596619094686

Offset: 0

Author

Paul D. Hanna, Jun 29 2022

Keywords

• nonn

Comments

a(n) = Sum_{k=0..n} A355350(n,k) * 3^k for n >= 0.

Examples

			G.f.: A(x) = 1 + 3*x + 18*x^2 + 108*x^3 + 660*x^4 + 4275*x^5 + 29106*x^6 + 205377*x^7 + 1485279*x^8 + 10943424*x^9 + 81866493*x^10 + ...
where
3*x = ... - x^10/A(x)^5 + 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 -+ ...
also,
3*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
where P(x) is the partition function and begins
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A355350, A355351, A355352, A355354, A355355, A355356, A355357.

Programs

PARI

{a(n) = my(A=[1,3],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = -polcoeff( sum(m=-t,t, (-1)^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) satisfies:

(1) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

(2) 3*x*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.