A183235 Sums of the cubes of multinomial coefficients.

1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, 75263273895385, 54867365927680618, 54868847079435960134, 73030508546599681432983, 126197144644287414997433576, 277255161467330877411064074059

Offset: 0

Author

Paul D. Hanna, Jan 04 2011

Keywords

• nonn

Comments

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

Ignoring initial term, equals the logarithmic derivative of A182963.

Examples

			G.f.: A(x) = 1 + x + 9*x^2/2!^3 + 244*x^3/3!^3 + 15833*x^4/4!^3 +...
A(x) = 1/((1-x)*(1-x^2/2!^3)*(1-x^3/3!^3)*(1-x^4/4!^3)*...).
...
After the initial term a(0)=1, the next few terms are
a(1) = 1^3 = 1,
a(2) = 1^3 + 2^3 = 9,
a(3) = 1^3 + 3^3 + 6^3 = 244,
a(4) = 1^3 + 4^3 + 6^3 + 12^3 + 24^3 = 15833,
a(5) = 1^3 + 5^3 + 10^3 + 20^3 + 30^3 + 60^3 + 120^3 = 1980126, ...;
and continue with the sums of cubes of the terms in triangle A036038.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..180

Crossrefs

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

Programs

PARI

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

Formula

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

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

Extensions

Examples added and name changed by Paul D. Hanna, Jan 05 2011