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A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.

%I A381676 #32 Mar 29 2025 04:26:48
%S A381676 1,1,4,17,86,472,2752,16753,105394,680366,4484360,30067160,204508240,
%T A381676 1408057120,9796738304,68786005361,486845236106,3470187822754,
%U A381676 24891491746792,179556655434382,1301857088258836,9482632068303296,69361538748381824,509303099950899352
%N A381676 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^2.
%H A381676 Vincenzo Librandi, <a href="/A381676/b381676.txt">Table of n, a(n) for n = 0..1000</a>
%F A381676 a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) ).
%F A381676 a(n) ~ 2^(3*n+3) / (Pi * 3^(3/2) * n^2). - _Vaclav Kotesovec_, Mar 26 2025
%t A381676 Table[Sum[Binomial[n,k]^2*(Binomial[n,k]-Binomial[n,k-1]),{k,0,n}],{n,0,20}] (* _Vincenzo Librandi_, Mar 27 2025 *)
%o A381676 (PARI) a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^2);
%o A381676 (Magma) [ &+[Binomial(n, k)^2 * (Binomial(n, k) - (k gt 0 select Binomial(n, k-1) else 0)) : k in [0..n]] : n in [0..20] ]; // _Vincenzo Librandi_, Mar 27 2025
%Y A381676 Cf. A000108, A129123, A382433, A382443, A382446.
%Y A381676 Cf. A131428, A178824.
%Y A381676 Cf. A000172, A086618, A238115.
%K A381676 nonn
%O A381676 0,3
%A A381676 _Seiichi Manyama_, Mar 25 2025