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Sums of the cubes of multinomial coefficients.
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%I #13 Feb 19 2015 16:17:36

%S 1,1,9,244,15833,1980126,428447592,146966837193,75263273895385,

%T 54867365927680618,54868847079435960134,73030508546599681432983,

%U 126197144644287414997433576,277255161467330877411064074059

%N Sums of the cubes of multinomial coefficients.

%C Equals sums of the cubes of terms in rows of the triangle of multinomial coefficients (A036038).

%C Ignoring initial term, equals the logarithmic derivative of A182963.

%H Vaclav Kotesovec, <a href="/A183235/b183235.txt">Table of n, a(n) for n = 0..180</a>

%F G.f.: Sum_{n>=0} a(n)*x^n/n!^3 = Product_{n>=1} 1/(1 - x^n/n!^3).

%F a(n) ~ c * (n!)^3, where c = Product_{k>=2} 1/(1-1/(k!)^3) = 1.14825648754771664323845829539510031170864046029463094659207423270573478812675... . - _Vaclav Kotesovec_, Feb 19 2015

%e G.f.: A(x) = 1 + x + 9*x^2/2!^3 + 244*x^3/3!^3 + 15833*x^4/4!^3 +...

%e A(x) = 1/((1-x)*(1-x^2/2!^3)*(1-x^3/3!^3)*(1-x^4/4!^3)*...).

%e ...

%e After the initial term a(0)=1, the next few terms are

%e a(1) = 1^3 = 1,

%e a(2) = 1^3 + 2^3 = 9,

%e a(3) = 1^3 + 3^3 + 6^3 = 244,

%e a(4) = 1^3 + 4^3 + 6^3 + 12^3 + 24^3 = 15833,

%e a(5) = 1^3 + 5^3 + 10^3 + 20^3 + 30^3 + 60^3 + 120^3 = 1980126, ...;

%e and continue with the sums of cubes of the terms in triangle A036038.

%o (PARI) {a(n)=n!^3*polcoeff(1/prod(k=1, n, 1-x^k/k!^3 +x*O(x^n)), n)}

%Y Cf. A036038, A005651, A183240, A183236, A183237, A183238; A182963.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 04 2011

%E Examples added and name changed by _Paul D. Hanna_, Jan 05 2011