A357799 a(n) = coefficient of x^n in A(x) such that: 1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * (A(x) + x^n)^(n+1).

1, 1, 4, 10, 33, 105, 363, 1268, 4600, 16954, 63663, 242180, 932255, 3623239, 14200924, 56061965, 222728379, 889828825, 3572675122, 14408128581, 58338540673, 237067134533, 966522205819, 3952323714926, 16206324436147, 66621153183615, 274505283101713

Offset

0,3

Links

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

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

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

a(n) ~ c * d^n / n^(3/2), where d = 4.3661068223321816847380533247926... and c = 0.8485754894190316922838386890774... - Vaclav Kotesovec, Jan 01 2023

Example

G.f.: A(x) = 1 + x + 4*x^2 + 10*x^3 + 33*x^4 + 105*x^5 + 363*x^6 + 1268*x^7 + 4600*x^8 + 16954*x^9 + 63663*x^10 + 242180*x^11 + 932255*x^12 + ...

Prog

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

Crossrefs

Sequence in context: A007716 A122948 A317800 * A149171 A149172 A338296

Adjacent sequences: A357796 A357797 A357798 * A357800 A357801 A357802

Keyword

nonn

Author

Paul D. Hanna, Dec 23 2022

Status

approved