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A183235 Sums of the cubes of multinomial coefficients. 8
1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, 75263273895385, 54867365927680618, 54868847079435960134, 73030508546599681432983, 126197144644287414997433576, 277255161467330877411064074059 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A112028 A368769 A329305 * A359732 A259673 A272234
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 04 2011
EXTENSIONS
Examples added and name changed by Paul D. Hanna, Jan 05 2011
STATUS
approved

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Last modified July 21 06:08 EDT 2024. Contains 374463 sequences. (Running on oeis4.)