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A348457 a(n) = Sum_{s=0..n} (-1)^s * ( Sum_{k=0..s} binomial(n,k) )^3.

%I A348457 #15 Nov 01 2021 04:03:27
%S A348457 1,-7,38,-232,1928,-16672,133508,-1044736,8337920,-67162624,537953708,
%T A348457 -4294193152,34336008272,-274889261056,2199555817952,-17592017354752,
%U A348457 140725278645248,-1125902437187584,9007484455265852,-72057555801604096,576453982622834768,-4611686599619510272,36893651043755447672,-295147896302964047872
%N A348457 a(n) = Sum_{s=0..n} (-1)^s * ( Sum_{k=0..s} binomial(n,k) )^3.
%H A348457 Seiichi Manyama, <a href="/A348457/b348457.txt">Table of n, a(n) for n = 0..1000</a>
%H A348457 Z. Zhang, <a href="https://doi.org/10.1016/S0012-365X(98)00178-2">A binomial identity related to Calkin's</a>, Discr. Math., 196 (1999), 287-289.
%F A348457 a(n) ~ (-1)^n * 2^(3*n-1). - _Vaclav Kotesovec_, Nov 01 2021
%p A348457 f := n -> add( (-1)^s*( add(binomial(n,k),k=0..s)^3 ),s=0..n);
%p A348457 [seq(f(n),n=0..50)];
%t A348457 a[n_] := Sum[(-1)^m * Sum[Binomial[n, k], {k, 0, m}]^3, {m, 0, n}]; Array[a, 24, 0] (* _Amiram Eldar_, Oct 28 2021 *)
%o A348457 (PARI) a(n) = sum(s=0, n, (-1)^s*sum(k=0, s, binomial(n,k))^3); \\ _Michel Marcus_, Oct 28 2021
%Y A348457 Cf. A007403.
%K A348457 sign
%O A348457 0,2
%A A348457 _N. J. A. Sloane_, Oct 28 2021