A183235 Sums of the cubes of multinomial coefficients.

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

Offset

0,3

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.

Links

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

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

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A112028 A368769 A329305 * A359732 A259673 A272234

Adjacent sequences: A183232 A183233 A183234 * A183236 A183237 A183238

Keyword

nonn

Author

Paul D. Hanna, Jan 04 2011

Extensions

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

Status

approved