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A382443 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^4.

%I A382443 #30 Mar 29 2025 16:25:59
%S A382443 1,1,4,65,566,10912,164032,3237313,62253130,1314421886,28392213224,
%T A382443 639799858304,14785604868256,350615631856960,8485316740880384,
%U A382443 209179475361783233,5239271305444731698,133100429387161703962,3424142506153260211720,89090362800169426107070
%N A382443 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^4.
%H A382443 Vincenzo Librandi, <a href="/A382443/b382443.txt">Table of n, a(n) for n = 0..500</a>
%F A382443 a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^3.
%F A382443 a(n) ~ 3 * 2^(5*n+6) / (Pi^2 * 5^(5/2) * n^4). - _Vaclav Kotesovec_, Mar 26 2025
%t A382443 Table[Sum[Binomial[n,k]*(Binomial[n,k]-Binomial[n,k-1])^4,{k,0,n}],{n,0,20}] (* _Vincenzo Librandi_, Mar 29 2025 *)
%o A382443 (PARI) a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^4);
%o A382443 (Magma) [&+[Binomial(n, k)* (Binomial(n, k) - Binomial (n, k-1))^4: k in [0..n]]: n in [0..21]]; // _Vincenzo Librandi_, Mar 29 2025
%Y A382443 Cf. A000108, A129123, A381676, A382433, A382446.
%Y A382443 Cf. A382434.
%K A382443 nonn
%O A382443 0,3
%A A382443 _Seiichi Manyama_, Mar 26 2025