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

0,3

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).

Links

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

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

Example

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.

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A295552 A066998 A036246 * A299715 A000872 A307406

Adjacent sequences: A183237 A183238 A183239 * A183241 A183242 A183243

Keyword

nonn

Author

Paul D. Hanna, Jan 03 2011

Extensions

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

Status

approved