A384824 G.f. A(x) satisfies 1/x^19 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+8).

1, 1, 5, 38, 319, 2871, 27507, 273925, 2808973, 29457644, 314470771, 3405995019, 37334767867, 413397265017, 4617060957512, 51951448775027, 588371324004508, 6701761863368579, 76723673176823126, 882342098781937683, 10188542630975395255, 118082022786322630334, 1373108879790849494070

Offset: 0

Author

Paul D. Hanna, Jun 10 2025

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 319*x^4 + 2871*x^5 + 27507*x^6 + 273925*x^7 + 2808973*x^8 + 29457644*x^9 + 314470771*x^10 + ...
		

Links

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

Crossrefs

Cf. A366731, A384821, A384822, A384823, A384825, A384826, A384827, A384828.

Programs

PARI

{a(n) = my(A=[1,1,0,0,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-4] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+8) ), #A-25)); 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^19 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+8).

(2) x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-4)*(n-5)) / (1 - x^n)^(n-8).

a(n) ~ c * d^n / n^(3/2), where d = 12.46033620173328231233579215988893957838459959... and c = 0.113752375605091798753361983956448030623... - Vaclav Kotesovec, Jun 11 2025