A215910 a(n) = sum of the n-th power of the multinomial coefficients in row n of triangle A036038.

1, 1, 5, 244, 354065, 25688403126, 141528428949437282, 83257152559805973052807833, 7012360438832401192319979008881500417, 109324223115831487504443410090345278639832867784010, 396327911646787133737309113762487915762995734538047874429637296650

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..30

Formula

a(n) = [x^n/n!^n] * Product_{k=1..n} 1/(1 - x^k/k!^n) for n>=1, with a(0)=1.

Logarithmic derivative of A215911, ignoring the initial term a(0).

a(n) ~ (n!)^n = A036740(n). - Vaclav Kotesovec, Feb 19 2015

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Feb 19 2015

Example

The sums of the n-th power of multinomial coefficients in row n of triangle A036038 begin:
a(1) = 1^1 = 1;
a(2) = 1^2 + 2^2 = 5;
a(3) = 1^3 + 3^3 + 6^3 = 244;
a(4) = 1^4 + 4^4 + 6^4 + 12^4 + 24^4 = 354065;
a(5) = 1^5 + 5^5 + 10^5 + 20^5 + 30^5 + 60^5 + 120^5 = 25688403126;
a(6) = 1^6 + 6^6 + 15^6 + 20^6 + 30^6 + 60^6 + 90^6 + 120^6 + 180^6 + 360^6 + 720^6 = 141528428949437282;
a(7) = 1^7 + 7^7 + 21^7 + 35^7 + 42^7 + 105^7 + 140^7 + 210^7 + 210^7 + 420^7 + 630^7 + 840^7 + 1260^7 + 2520^7 + 5040^7 = 83257152559805973052807833; ...
which also form a logarithmic generating function of an integer series:
L(x) = x + 5*x^2/2 + 244*x^3/3 + 354065*x^4/4 + 25688403126*x^5/5 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 84*x^3 + 88602*x^4 + 5137769389*x^5 +...+ A215911(n)*x^n +...

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
    end:
a:= n-> n!^n*b(n$3):
seq(a(n), n=0..12);  # Alois P. Heinz, Sep 11 2019

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, 1, b[n - i, Min[n - i, i], k]/i!^k + b[n, i - 1, k]];
a[n_] := n!^n b[n, n, n];
a /@ Range[0, 12] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

Prog

(PARI) {a(n)=n!^n*polcoeff(1/prod(m=1, n, 1-x^m/m!^n +x*O(x^n)), n)}
for(n=1, 15, print1(a(n), ", "))

Crossrefs

Cf. A183610, A036038, A036740, A005651, A183240, A183235, A183236, A183237, A183238; A215911.

Cf. A326321.

Sequence in context: A327582 A144999 A361538 * A097323 A166943 A125533

Adjacent sequences: A215907 A215908 A215909 * A215911 A215912 A215913

Keyword

nonn

Author

Paul D. Hanna, Aug 26 2012

Status

approved