A183240 Sums of the squares of multinomial coefficients.

1, 1, 5, 46, 773, 19426, 708062, 34740805, 2230260741, 180713279386, 18085215373130, 2188499311357525, 315204533416762046, 53270712928769375885, 10441561861586014363349, 2349364090881443819316871, 601444438364480313663234821, 173817677082622796179263021770

Offset: 0

Author

Paul D. Hanna, Jan 03 2011

Keywords

• nonn

Comments

Equals sums of the squares of terms in rows of the triangle of multinomial coefficients (A036038).

Ignoring initial term, equals the logarithmic derivative of A183241; A183241 is conjectured to consist entirely of integers.

More generally, let {M(n,k), n>=0} be the sums of the k-th powers of the multinomial coefficients where k>=0 (see table A183610), then:

Sum_{n>=0} M(n,k)*x^n/n!^k = Product_{n>=1} 1/(1-x^n/n!^k).

Examples

			G.f.: A(x) = 1 + x + 5*x^2/2!^2 + 46*x^3/3!^2 + 773*x^4/4!^2 +...
A(x) = 1/((1-x)*(1-x^2/2!^2)*(1-x^3/3!^2)*(1-x^4/4!^2)*...).
...
After the initial term a(0)=1, the next several terms are
a(1) = 1^2 = 1,
a(2) = 1^2 + 2^2 = 5,
a(3) = 1^2 + 3^2 + 6^2 = 46,
a(4) = 1^2 + 4^2 + 6^2 + 12^2 + 24^2 = 773,
a(5) = 1^2 + 5^2 + 10^2 + 20^2 + 30^2 + 60^2 + 120^2 = 19426,
and continue with the sums of squares of the terms in triangle A036038.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A036038, A005651, A183235, A183236, A183237, A183238; A183241.

Cf. A183610 (table of sums of powers of multinomial coefficients).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i))/i!^2+b(n, i-1))
    end:
a:= n-> n!^2*b(n$2):
seq(a(n), n=0..21);  # Alois P. Heinz, Sep 11 2019

Mathematica

t=Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i], Length[#1] > Length[#2] &]], {1}], {i, 30}]^2; Plus@@@t (* From Tony D. Noe *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1,
     b[n - i, Min[n - i, i]]/i!^2 + b[n, i - 1]];
a[n_] := n!^2 b[n, n];
a /@ Range[0, 21] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)

PARI

{a(n)=n!^2*polcoeff(1/prod(k=1,n,1-x^k/k!^2 +x*O(x^n)),n)}

Formula

G.f.: Sum_{n>=0} a(n)*x^n/n!^2 = Product_{n>=1} 1/(1-x^n/n!^2).

a(n) ~ c * (n!)^2, where c = Product_{k>=2} 1/(1-1/(k!)^2) = 1.37391178018464563291052028168404977854977270679629932106310942272080844... . - Vaclav Kotesovec, Feb 19 2015

Extensions

Terms following a(7) computed by T. D. Noe.