A371684 a(n) = Sum_{k=0..n} 2^(3*k)*binomial(2*n, 2*k)*Euler(2*k, 1/2). Alternating row sums of A371637.

1, -1, 9, -217, 9841, -717841, 76804665, -11330490025, 2204195526241, -546715992537505, 168397490614671849, -63062013420332052985, 28216110792407667898321, -14866226664969958126495921, 9109882748673411939937074969, -6424247756451800785395922510537

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Formula

a(n) ~ (-1)^n * cos(Pi/(2*sqrt(2))) * 2^(5*n+3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)). - Vaclav Kotesovec, Apr 03 2024

From Seiichi Manyama, Apr 18 2026: (Start)

a(n) = (2*n)! * [x^(2*n)] cosh(x) / cosh(sqrt(2)*x).

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

Maple

seq(add(2^(3*k)*binomial(2*n, 2*k)*euler(2*k, 1/2), k = 0..n), n = 0..15);

Mathematica

Table[Sum[2^(3*k)*Binomial[2*n, 2*k]*EulerE[2*k, 1/2], {k, 0, n}], {n, 0, 20}] (* Paolo Xausa, Apr 17 2024 *)

Prog

(PARI) a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1-sum(j=0, i-1, 2^(i-j)*binomial(2*i, 2*j)*v[j+1])); v; \\ Seiichi Manyama, Apr 18 2026

Crossrefs

Cf. A371637, A371683.

Sequence in context: A382842 A197669 A368632 * A393754 A157692 A299548

Adjacent sequences: A371681 A371682 A371683 * A371685 A371686 A371687

Keyword

sign

Author

Peter Luschny, Apr 03 2024

Status

approved