A381341 Expansion of e.g.f. exp( x * cosh(sqrt(2)*x) ).

1, 1, 1, 7, 25, 81, 601, 3207, 18705, 156385, 1087441, 8962823, 84001897, 732712241, 7487525865, 78537490951, 831722893217, 9804469109953, 115549730623009, 1431784628480007, 18795444460125241, 248964703826005777, 3487888859183694329, 50283005924345951111

Offset: 0

Author

Seiichi Manyama, Feb 20 2025

Keywords

• nonn

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A003727, A009189, A381273, A381274, A381275, A381276, A381342.

Cf. A185951.

Programs

PARI

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, 2^((n-k)/2)*a185951(n, k));

Formula

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

a(n) = Sum_{k=0..n} 2^((n-k)/2) * A185951(n,k).