A381284 Expansion of e.g.f. 1/(1 - sinh(3*x) / 3).

1, 1, 2, 15, 96, 741, 7632, 87795, 1149696, 17155881, 282880512, 5128464375, 101592631296, 2178698451021, 50314379323392, 1245198047833755, 32868161979088896, 921803465256094161, 27373850876851126272, 858044392807801699935, 28311289100161039466496

Offset: 0

Author

Seiichi Manyama, Feb 18 2025

Keywords

• nonn

Crossrefs

Cf. A006154, A191277.

Cf. A136630.

Programs

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*3^(n-k)*a136630(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 9^k * binomial(n,2*k+1) * a(n-2*k-1).

a(n) = Sum_{k=0..n} k! * 3^(n-k) * A136630(n,k).

a(n) ~ sqrt(Pi/5) * 3^(n+1) * n^(n + 1/2) / (arcsinh(3)^(n+1) * exp(n)). - Vaclav Kotesovec, Apr 19 2025