A369084 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^(n*(n-1)/2) * (x^n - A(x))^n.

1, 1, 1, 1, 3, 3, 8, 12, 29, 48, 105, 202, 420, 831, 1729, 3538, 7370, 15293, 32094, 67410, 142221, 301074, 639076, 1360991, 2903607, 6213695, 13318015, 28616357, 61576994, 132779990, 286704638, 620144700, 1343082108, 2913091456, 6325803831, 13754042495, 29937461161

Offset

1,5

Links

Paul D. Hanna, Table of n, a(n) for n = 1..600

Formula

G.f. A(x) satisfies:

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

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

Example

G.f.: A(x) = x + x^2 + x^3 + x^4 + 3*x^5 + 3*x^6 + 8*x^7 + 12*x^8 + 29*x^9 + 48*x^10 + 105*x^11 + 202*x^12 + 420*x^13 + 831*x^14 + 1729*x^15 + ...
where
1 = ... + x^6/(1/x^3 - A(x))^3 + x^3/(1/x^2 - A(x))^2 + x/(1/x - A(x)) + 1 + (x - A(x)) + x*(x^2 - A(x))^2 + x^3*(x^3 - A(x))^3 + x^6*(x^4 - A(x))^4 + ... + x^(n*(n-1)/2)*(x^n - A(x))^n + ...

Prog

(PARI) {a(n) = my(A=[0], M); for(i=1, n, A=concat(A, 0); M = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff(-1 + sum(m=-M, M, x^(m*(m-1)/2) * (x^m - Ser(A))^m ), #A-1)); A[n+1]}
for(n=1, 40, print1(a(n), ", "))

Crossrefs

Cf. A355861.

Sequence in context: A364468 A303902 A090597 * A304887 A126073 A126592

Adjacent sequences: A369081 A369082 A369083 * A369085 A369086 A369087

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2024

Status

approved