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A382435 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^6.

%I A382435 #15 Mar 25 2025 12:56:36
%S A382435 1,1,3,129,1587,39443,1125383,30211457,1107074979,36214609683,
%T A382435 1433494688871,54495716261011,2275005440977063,95146470595975399,
%U A382435 4170974287982618639,185640304224109725569,8492643748223480148419,395051289603660979274339,18726850582009755291702599
%N A382435 a(n) = Sum_{k=0..n} ( binomial(n,k) - binomial(n,k-1) )^6.
%F A382435 a(n) = Sum_{k=0..n} A080233(n,k)^6 = Sum_{k=0..n} A156644(n,k)^6.
%F A382435 a(n) = 2 * A382433(n) - 1.
%p A382435 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A382435      `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
%p A382435     end:
%p A382435 a:= n-> 2*add(b(n, n-2*j)^6, j=0..n/2)-1:
%p A382435 seq(a(n), n=0..18);  # _Alois P. Heinz_, Mar 25 2025
%o A382435 (PARI) a(n) = sum(k=0, n, (binomial(n, k)-binomial(n, k-1))^6);
%o A382435 (Python)
%o A382435 from math import comb
%o A382435 def A382435(n): return (sum((comb(n,j)*(m:=n-(j<<1)+1)//(m+j))**6 for j in range((n>>1)+1))<<1)-1 # _Chai Wah Wu_, Mar 25 2025
%Y A382435 Cf. A131428, A382434.
%Y A382435 Cf. A080233, A156644, A382433.
%K A382435 nonn
%O A382435 0,3
%A A382435 _Seiichi Manyama_, Mar 25 2025