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%I #30 Jun 04 2021 03:17:26
%S 1,1,5,46,773,19426,708062,34740805,2230260741,180713279386,
%T 18085215373130,2188499311357525,315204533416762046,
%U 53270712928769375885,10441561861586014363349,2349364090881443819316871,601444438364480313663234821,173817677082622796179263021770
%N Sums of the squares of multinomial coefficients.
%C Equals sums of the squares of terms in rows of the triangle of multinomial coefficients (A036038).
%C Ignoring initial term, equals the logarithmic derivative of A183241; A183241 is conjectured to consist entirely of integers.
%C 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:
%C Sum_{n>=0} M(n,k)*x^n/n!^k = Product_{n>=1} 1/(1-x^n/n!^k).
%H Vaclav Kotesovec, <a href="/A183240/b183240.txt">Table of n, a(n) for n = 0..250</a>
%F G.f.: Sum_{n>=0} a(n)*x^n/n!^2 = Product_{n>=1} 1/(1-x^n/n!^2).
%F a(n) ~ c * (n!)^2, where c = Product_{k>=2} 1/(1-1/(k!)^2) = 1.37391178018464563291052028168404977854977270679629932106310942272080844... . - _Vaclav Kotesovec_, Feb 19 2015
%e G.f.: A(x) = 1 + x + 5*x^2/2!^2 + 46*x^3/3!^2 + 773*x^4/4!^2 +...
%e A(x) = 1/((1-x)*(1-x^2/2!^2)*(1-x^3/3!^2)*(1-x^4/4!^2)*...).
%e ...
%e After the initial term a(0)=1, the next several terms are
%e a(1) = 1^2 = 1,
%e a(2) = 1^2 + 2^2 = 5,
%e a(3) = 1^2 + 3^2 + 6^2 = 46,
%e a(4) = 1^2 + 4^2 + 6^2 + 12^2 + 24^2 = 773,
%e a(5) = 1^2 + 5^2 + 10^2 + 20^2 + 30^2 + 60^2 + 120^2 = 19426,
%e and continue with the sums of squares of the terms in triangle A036038.
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p b(n-i, min(n-i, i))/i!^2+b(n, i-1))
%p end:
%p a:= n-> n!^2*b(n$2):
%p seq(a(n), n=0..21); # _Alois P. Heinz_, Sep 11 2019
%t t=Table[Apply[Multinomial, Reverse[Sort[IntegerPartitions[i], Length[#1] > Length[#2] &]], {1}], {i, 30}]^2; Plus@@@t (* From Tony D. Noe *)
%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1,
%t b[n - i, Min[n - i, i]]/i!^2 + b[n, i - 1]];
%t a[n_] := n!^2 b[n, n];
%t a /@ Range[0, 21] (* _Jean-François Alcover_, Jun 04 2021, after _Alois P. Heinz_ *)
%o (PARI) {a(n)=n!^2*polcoeff(1/prod(k=1,n,1-x^k/k!^2 +x*O(x^n)),n)}
%Y Cf. A036038, A005651, A183235, A183236, A183237, A183238; A183241.
%Y Cf. A183610 (table of sums of powers of multinomial coefficients).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 03 2011
%E Terms following a(7) computed by _T. D. Noe_.
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