A384827 G.f. A(x) satisfies -1/x^55 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+14).

1, 1, 8, 95, 1288, 19116, 300511, 4918268, 82918049, 1430142380, 25115651237, 447578072658, 8073426806649, 147122009148252, 2704441907759235, 50088849266618466, 933792151007378231, 17509062834076661230, 329985690688947517626, 6247533413700369107192, 118768564127167799819733

Offset: 0

Author

Paul D. Hanna, Jun 10 2025

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 95*x^3 + 1288*x^4 + 19116*x^5 + 300511*x^6 + 4918268*x^7 + 82918049*x^8 + 1430142380*x^9 + 25115651237*x^10 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A366731, A384821, A384822, A384823, A384824, A384825, A384826, A384828.

Programs

PARI

{a(n) = my(A=[1,1,0,0,0,0,0,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-7] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+14) ), #A-64)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

(1) -1/x^55 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+14).

(2) -x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-7)*(n-8)) / (1 - x^n)^(n-14).

a(n) ~ c * d^n / n^(3/2), where d = 20.5190724870235230419993391970202418416256614273528... and c = 0.06135641554365612129872100433075021800206923942... - Vaclav Kotesovec, Jun 11 2025