A355354 G.f. A(x) satisfies: 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

1, 4, 28, 196, 1416, 10860, 87392, 727188, 6196212, 53783336, 474011756, 4231158016, 38174676188, 347566170384, 3189295781780, 29465038957708, 273851282010308, 2558703740102840, 24019990008557160, 226444571054525156, 2142925363606256584, 20349477565111498148

Offset

0,2

Comments

a(n) = Sum_{k=0..n} A355350(n,k) * 4^k for n >= 0.

Links

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

Formula

G.f. A(x) satisfies:

(1) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

(2) 4*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 + 4*x + 28*x^2 + 196*x^3 + 1416*x^4 + 10860*x^5 + 87392*x^6 + 727188*x^7 + 6196212*x^8 + 53783336*x^9 + 474011756*x^10 + ...
where
4*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,
4*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, 4], 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, A355355, A355356, A355357.

Sequence in context: A002903 A208704 A270471 * A019482 A198630 A246021

Adjacent sequences: A355351 A355352 A355353 * A355355 A355356 A355357

Keyword

nonn

Author

Paul D. Hanna, Jun 29 2022

Status

approved