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G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^4 * x^k*(1-x)^(n-k).
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%I #5 Oct 23 2012 15:01:50

%S 1,1,1,15,65,175,1155,7229,31081,162755,1018195,5448645,28879019,

%T 171229541,991796825,5540516759,32232757569,190448140543,

%U 1105001828111,6463178610505,38380301015595,227218808375165,1344777832158545,8022879439995215,47988823484272835

%N G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k)^4 * x^k*(1-x)^(n-k).

%e G.f.: A(x) = 1 + x + x^2 + 15*x^3 + 65*x^4 + 175*x^5 + 1155*x^6 + 7229*x^7 +...

%e where

%e A(x) = 1 +

%e x*((1-x) + x) +

%e x^2*((1-x)^2 + 2^4*x*(1-x) + x^2) +

%e x^3*((1-x)^3 + 3^4*x*(1-x)^2 + 3^4*x^2*(1-x) + x^3) +

%e x^4*((1-x)^4 + 4^4*x*(1-x)^3 + 6^4*x^2*(1-x)^2 + 4^4*x^3*(1-x) + x^4) +

%e x^5*((1-x)^5 + 5^4*x*(1-x)^4 + 10^4*x^2*(1-x)^3 + 10^4*x^3*(1-x)^2 + 5^4*x^4*(1-x) + x^5) +...

%o (PARI) {a(n)=polcoeff(sum(m=0, n+1, x^m*sum(k=0, m, binomial(m, k)^4*x^k*(1-x)^(m-k) + x*O(x^n))), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A217615, A217421.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Oct 23 2012