cp's OEIS Frontend

A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.

%I A382433 #19 Apr 01 2025 20:09:29
%S A382433 1,1,2,65,794,19722,562692,15105729,553537490,18107304842,
%T A382433 716747344436,27247858130506,1137502720488532,47573235297987700,
%U A382433 2085487143991309320,92820152112054862785,4246321874111740074210,197525644801830489637170,9363425291004877645851300
%N A382433 a(n) = S(6,n), where S(r,n) = Sum_{k=0..floor(n/2)} ( binomial(n,k) - binomial(n,k-1) )^r.
%H A382433 Alois P. Heinz, <a href="/A382433/b382433.txt">Table of n, a(n) for n = 0..562</a>
%F A382433 a(n) = Sum_{k=0..floor(n/2)} A008315(n,k)^6.
%F A382433 a(n) = Sum_{k=0..n} A120730(n,k)^6.
%F A382433 a(n) = A357824(n,6).
%F A382433 a(n) = Sum_{k=0..n} binomial(n,k) * ( binomial(n,k) - binomial(n,k-1) )^5.
%F A382433 a(n) ~ 5 * 2^(6*n+4) / (3^(5/2) * Pi^(5/2) * n^(11/2)). - _Vaclav Kotesovec_, Mar 25 2025
%p A382433 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A382433      `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
%p A382433     end:
%p A382433 a:= n-> add(b(n, n-2*j)^6, j=0..n/2):
%p A382433 seq(a(n), n=0..18);  # _Alois P. Heinz_, Mar 25 2025
%t A382433 Table[Sum[Binomial[n,k] * (Binomial[n,k] - Binomial[n,k-1])^5, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Mar 25 2025 *)
%o A382433 (PARI) a(n) = sum(k=0, n, binomial(n, k)*(binomial(n, k)-binomial(n, k-1))^5);
%o A382433 (Python)
%o A382433 from math import comb
%o A382433 def A382433(n): return sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1)) # _Chai Wah Wu_, Mar 25 2025
%Y A382433 Column k=6 of A357824.
%Y A382433 Cf. A000108, A129123.
%Y A382433 Cf. A008315, A120730, A382435.
%K A382433 nonn
%O A382433 0,3
%A A382433 _Seiichi Manyama_, Mar 25 2025