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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 03 2011
EXTENSIONS
Terms following a(7) computed by T. D. Noe.
STATUS
approved