A355363 G.f. A(x) satisfies: 3*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 3, 27, 270, 2928, 33912, 411345, 5159337, 66364326, 870637086, 11604385575, 156697653654, 2139109221960, 29472597414681, 409312118499336, 5723853297702444, 80528723782556475, 1139033786793330429, 16187921479930951917, 231046413762053945958

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0..n} A355360(n,k) * 3^k for n >= 0.

G.f. A(x) satisfies:

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

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

(3) 3*x*A(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.

Example

G.f.: A(x) = 1 + 3*x + 27*x^2 + 270*x^3 + 2928*x^4 + 33912*x^5 + 411345*x^6 + 5159337*x^7 + 66364326*x^8 + 870637086*x^9 + 11604385575*x^10 + ...
where
3*x*A(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 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...

Prog

(PARI) {a(n) = my(A=[1, 3], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+9));
A[#A] = polcoeff( 3*x*Ser(A) - 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), ", "))

Crossrefs

Cf. A355360, A355361, A355362, A355364, A355365.

Sequence in context: A199688 A275998 A098401 * A011544 A127503 A214363

Adjacent sequences: A355360 A355361 A355362 * A355364 A355365 A355366

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2022

Status

approved