A355355 G.f. A(x) satisfies: 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 5, 40, 320, 2660, 23455, 216540, 2064055, 20137945, 200134600, 2019406895, 20635313325, 213109960895, 2220820915065, 23323755734820, 246616999661690, 2623193780773530, 28049464032800110, 301340494687086960, 3251017466141039095, 35207152686408604400

Offset

0,2

Comments

a(n) = Sum_{k=0..n} A355350(n,k) * 5^k for n >= 0.

Links

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

Formula

G.f. A(x) satisfies:

(1) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

(2) 5*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 + 5*x + 40*x^2 + 320*x^3 + 2660*x^4 + 23455*x^5 + 216540*x^6 + 2064055*x^7 + 20137945*x^8 + 200134600*x^9 + 2019406895*x^10 + ...
where
5*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,
5*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 + ...

Prog

(PARI) {a(n) = my(A=[1, 5], 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), ", "))

Crossrefs

Cf. A355350, A355351, A355352, A355353, A355354, A355356, A355357.

Sequence in context: A145841 A123943 A067412 * A078846 A027259 A007036

Adjacent sequences: A355352 A355353 A355354 * A355356 A355357 A355358

Keyword

nonn

Author

Paul D. Hanna, Jun 29 2022

Status

approved