A384168 a(n) = 3^n * n! * binomial(4*n/3,n) * Sum_{k=1..n} 1/(n+3*k).

1, 13, 234, 5566, 165944, 5966136, 251491120, 12169996912, 665146831680, 40530954643840, 2724842629685120, 200361647815660800, 15997170878205905920, 1378271357428552115200, 127459020533529062246400, 12593128815600367187507200, 1323895109721239722075136000

Offset

1,2

Links

Table of n, a(n) for n=1..17.

Formula

a(n) = Sum_{k=0..n} k * (n+3)^(k-1) * 3^(n-k) * |Stirling1(n,k)|.

a(n) = n! * [x^n] ( -log(1 - 3*x)/(3 * (1 - 3*x)^(n/3+1)) ).

a(n) = 3^(n-1)*binomial(4*n/3, n)*n!*(PolyGamma(0, 1+4*n/3) - PolyGamma(0, 1+n/3)). - Stefano Spezia, Sep 19 2025

a(n) ~ log(2) * 2^(8*n/3+2) * (n/e)^n / 3. - Amiram Eldar, Nov 07 2025

Mathematica

a[n_]:=3^n * n! * Binomial[4*n/3, n] * Sum[1/(n+3*k), {k, n}]; Array[a, 17] (* Stefano Spezia, Sep 19 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, k*(n+3)^(k-1)*3^(n-k)*abs(stirling(n, k, 1)));

Crossrefs

Cf. A098118, A384167, A384169.

Cf. A303486.

Sequence in context: A218202 A134493 A051689 * A000824 A229384 A181035

Adjacent sequences: A384165 A384166 A384167 * A384169 A384170 A384171

Keyword

nonn

Author

Seiichi Manyama, May 21 2025

Status

approved