A381744 Expansion of exp( Sum_{k>=1} binomial(6*k-1,2*k) * x^k/k ).

1, 10, 215, 5942, 186111, 6283192, 222992692, 8201608382, 309834609743, 11950890428170, 468707758663887, 18634632264615272, 749325132218313540, 30422303269317412048, 1245346665979469486376, 51343805279989437688334, 2130090659402456357279919, 88858984785475871013971710

Offset: 0

Author

Seiichi Manyama, Mar 05 2025

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..605

Crossrefs

Cf. A079489, A381745, A381746.

Cf. A006013, A182960.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(6*k-1, 2*k)*x^k/k)))

Formula

G.f. A(x) satisfies A(x^2) = B(x) * B(-x), where B(x) is the g.f. of A006013.

a(n) = Sum_{k=0..2*n} (-1)^k * A006013(k) * A006013(2*n-k).

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(6*k-1,2*k) * a(n-k).

G.f.: B(x)^2, where B(x) is the g.f. of A182960.