A182963 -id:A182963 - cp's OEIS frontend

A183235 Sums of the cubes of multinomial coefficients.

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

Offset: 0

Paul D. Hanna, Jan 04 2011

nonn

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.

			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.

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

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

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

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

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

A215911 G.f.: exp( Sum_{n>=1} A215910(n)*x^n/n ), where A215910(n) equals the sum of the n-th power of multinomial coefficients in row n of triangle A036038.

Original entry on oeis.org

1, 1, 3, 84, 88602, 5137769389, 23588076629522583, 11893878960703225919597767, 876545054865944028047877165082786426, 12147135901759930712215268630715086378214795245696, 39632791164678725520866813137932593902239710762044280903318659253

Offset: 0

Paul D. Hanna, Aug 26 2012

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 84*x^3 + 88602*x^4 + 5137769389*x^5 +...
such that the logarithm of the g.f. begins:
log(A(x)) = x + 5*x^2/2 + 244*x^3/3 + 354065*x^4/4 + 25688403126*x^5/5 + 141528428949437282*x^6/6 +...+ A215910(n)*x^n/n +...
where the coefficients A215910(n) begin:
A215910(1) = 1^1 = 1;
A215910(2) = 1^2 + 2^2 = 5;
A215910(3) = 1^3 + 3^3 + 6^3 = 244;
A215910(4) = 1^4 + 4^4 + 6^4 + 12^4 + 24^4 = 354065;
A215910(5) = 1^5 + 5^5 + 10^5 + 20^5 + 30^5 + 60^5 + 120^5 = 25688403126; ...
and equal the sums of the n-th power of multinomial coefficients in row n of triangle A036038.

Vaclav Kotesovec, Table of n, a(n) for n = 0..30

Cf. A215910, A183239, A183241, A183241, A182963.

{a(n)=local(L=sum(m=1,n,m!^m*polcoeff(1/prod(k=1, n, 1-x^k/k!^m +x*O(x^m)), m)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}
for(n=0,15,print1(a(n),", "))

a(n) ~ (n!)^n / n. - Vaclav Kotesovec, Feb 19 2015

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2 - 1) / exp(n^2 - 1/12). - Vaclav Kotesovec, Feb 19 2015

cp's OEIS Frontend

A183235 Sums of the cubes of multinomial coefficients.

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