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A382446 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^6.

%I A382446 #19 Mar 30 2025 09:52:53
%S A382446 1,1,4,257,4286,258952,11816512,632854273,43732565914,2637804065366,
%T A382446 207379028199080,14568483339859880,1205457271871693920,
%U A382446 95108827011788280160,8187664948710535579904,698818327346476962092801,62477582066507173352034866,5627626080883126186936773514
%N A382446 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^6.
%H A382446 Vincenzo Librandi, <a href="/A382446/b382446.txt">Table of n, a(n) for n = 0..400</a>
%F A382446 a(n) = Sum_{k=0..n} binomial(n,k)^2 * ( binomial(n,k) - binomial(n,k-1) )^5.
%F A382446 a(n) ~ 15 * 2^(7*n+9) / (Pi^3 * 7^(7/2) * n^6). - _Vaclav Kotesovec_, Mar 26 2025
%t A382446 Table[Sum[Binomial[n,k]*(Binomial[n,k]-Binomial[n,k-1])^6,{k,0,n}],{n,0,20}] (* _Vincenzo Librandi_, Mar 30 2025 *)
%o A382446 (PARI) a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^6);
%o A382446 (Magma) [&+[Binomial(n, k)* (Binomial(n, k) - Binomial (n, k-1))^6: k in [0..n]]: n in [0..21]]; // _Vincenzo Librandi_, Mar 30 2025
%Y A382446 Cf. A000108, A129123, A381676, A382433, A382443.
%Y A382446 Cf. A382435.
%K A382446 nonn
%O A382446 0,3
%A A382446 _Seiichi Manyama_, Mar 26 2025