A100872 a(n) = (1/sqrt(5)) * Sum_{k>0} k^(2n)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.

1, 13, 421, 25453, 2473141, 352444093, 69251478661, 17943523153933, 5927841361456981, 2431910546406522973, 1212989379862721528101, 722875495525684291639213, 507275965883448333971692021, 414031618935013558427928710653, 388884101194230308462039862028741

Offset

1,2

Comments

A bisection of "Stirling-Bernoulli transform" of Fibonacci numbers.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..223

Formula

a(n) = A050946(2*n).

From Peter Bala, Aug 20 2014: (Start)

E.g.f.: -1/2 + (1/2)*exp(z)/(3*exp(z) - exp(2*z) - 1) = z^2/2! + 13*z^4/4! + 421*z^6/6! + ....

a(n) = Sum_{k = 1..n} 2^(k-1)*A241171(n,k).

a(n) = Sum_{1 <= j <= k <= n} (-1)^(k-j)*binomial(2*k,k+j)*j^(2*n).

(End)

Mathematica

FullSimplify[Table[PolyLog[-2k, GoldenRatio^(-2)]/Sqrt[5], {k, 1, 10}]] (* Vladimir Reshetnikov, Feb 16 2011 *)
T[n_, k_] /; 1 <= k <= n := T[n, k] = k(2k-1) T[n-1, k-1] + k^2 T[n-1, k]; T[_, 1] = 1; T[_, _] = 0; a[n_] := Sum[2^(k-1) T[n, k], {k, 1, n}]; Array[a, 15] (* Jean-François Alcover, Jul 03 2019 *)

Prog

(PARI) a(n)=round(1/sqrt(5)*sum(k=1, 500, k^(2*n)/((1+sqrt(5))/2)^(2*k)))

Crossrefs

Cf. A001622, A050946, A100868, A241171.

Sequence in context: A098890 A012023 A081442 * A012045 A012109 A012084

Adjacent sequences: A100869 A100870 A100871 * A100873 A100874 A100875

Keyword

nonn,easy

Author

Benoit Cloitre, Jan 08 2005

Status

approved