login
Sums of multinomial coefficients to the 5th power.
7

%I #12 Mar 13 2015 22:53:52

%S 1,1,33,8020,8220257,25688403126,199758931567152,3357348771315829641,

%T 110013706232123658318433,6496199364012472451887572970,

%U 649619955166586474874295658148158,104621943411970982740307507415589286391

%N Sums of multinomial coefficients to the 5th power.

%C Equals sums of the 5th power of terms in rows of the triangle of multinomial coefficients (A036038).

%H Vaclav Kotesovec, <a href="/A183237/b183237.txt">Table of n, a(n) for n = 0..120</a>

%F G.f.: Sum_{n>=0} a(n)*x^n/n!^5 = Product_{n>=1} 1/(1 - x^n/n!^5).

%F a(n) ~ c * (n!)^5, where c = Product_{k>=2} 1/(1-1/(k!)^5) = 1.03239096052278897179685563337623849923796538921602982416328969955606263213989... . - _Vaclav Kotesovec_, Feb 19 2015

%e G.f.: A(x) = 1 + x + 33*x^2/2!^5 + 8020*x^3/3!^5 + 8220257*x^4/4!^5 +...

%e A(x) = 1/((1-x)*(1-x^2/2!^5)*(1-x^3/3!^5)*(1-x^4/4!^5)*...).

%o (PARI) {a(n)=n!^5*polcoeff(1/prod(k=1, n, 1-x^k/k!^5 +x*O(x^n)), n)}

%Y Cf. A036038, A005651, A183240, A183235, A183236, A183238.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 04 2011