A384822 G.f. A(x) satisfies 1/x^5 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+4).

1, 1, 5, 19, 109, 598, 3592, 22110, 140467, 911136, 6014277, 40260501, 272682397, 1865181921, 12866239311, 89403333632, 625211046931, 4396844409898, 31075863324446, 220618909826500, 1572549447431889, 11249693613964519, 80743512234554655, 581272589032594530, 4196118995069449989

Offset: 0

Author

Paul D. Hanna, Jun 10 2025

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 19*x^3 + 109*x^4 + 598*x^5 + 3592*x^6 + 22110*x^7 + 140467*x^8 + 911136*x^9 + 6014277*x^10 + ...
		

Links

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

Crossrefs

Cf. A366731, A384821, A384823, A384824, A384825, A384826, A384827, A384828.

Programs

PARI

{a(n) = my(A=[1,1,0,0]); for(i=1, n, A = concat(A, 0);
A[#A-2] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+4) ), #A-9)); 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^5 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+4).

(2) x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-2)*(n-3)) / (1 - x^n)^(n-4).

a(n) ~ c * d^n / n^(3/2), where d = 7.687452504111926947946743863677977... and c = 0.2779337748307189293421456249838... - Vaclav Kotesovec, Jun 11 2025