A215515 2n-th derivative of cos(x)^cos(x) at x=0.

1, -1, 7, -76, 1597, -41776, 1673167, -74527636, 4832747017, -305644428256, 30618856073947, -2276081971574236, 390042814538656957, -20435946140834126176, 10544180964356207226247, 604793906292405974180324, 688972694565220644332739217

Offset: 0

Author

Michel Lagneau, Aug 14 2012

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..243

Crossrefs

Cf. A000182, A215518, A354520.

Programs

Maple

a:= n-> coeff(series(cos(x)^cos(x), x, 2*n+1), x, 2*n)*(2*n)!:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 17 2023

Mathematica

f[x_] := Cos[x]^Cos[x]; Table[Derivative[2*n][f][0],{n,0,30}]

PARI

a354520(n) = sum(k=1, n\2, (16^k-4^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (-1)^j*a354520(2*j)*binomial(2*i-1, 2*j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Aug 17 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * A354520(2*k) * binomial(2*n-1,2*k-1) * a(n-k). - Seiichi Manyama, Aug 17 2022