|
%I #29 Jun 07 2019 11:32:35
%S 1,17,338,6754,131428,2495906,46434532,849488292,15328171208,
%T 273445276258,4831735919236,84688295720132,1474133269832776,
%U 25506505928857892,439034457665156168,7522356118216054216,128364598453699389840,2182553210810903666402,36989251585608710893636
%N a(n) = Sum_{m=0..n} (Sum_{k=0..m} binomial(n,k))^4.
%H Seiichi Manyama, <a href="/A294435/b294435.txt">Table of n, a(n) for n = 0..300</a>
%H N. J. Calkin, <a href="http://dx.doi.org/10.1016/0012-365X(94)90394-8">A curious binomial identity</a>, Discr. Math., 131 (1994), 335-337.
%H M. Hirschhorn, <a href="http://dx.doi.org/10.1016/0012-365X(95)00086-C">Calkin's binomial identity</a>, Discr. Math., 159 (1996), 273-278.
%F a(n) ~ n * 2^(4*n - 1). - _Vaclav Kotesovec_, Jun 07 2019
%p A:=proc(n,k) local j; add(binomial(n,j),j=0..k); end;
%p S:=proc(n,p) local i; global A; add(A(n,i)^p, i=0..n); end;
%p [seq(S(n,4),n=0..30)];
%t Table[Sum[Sum[Binomial[n,k], {k,0,m}]^4, {m,0,n}], {n,0,15}] (* _Vaclav Kotesovec_, Jun 07 2019 *)
%o (PARI) a(n) = sum(m=0, n, sum(k=0, m, binomial(n,k))^4); \\ _Michel Marcus_, Nov 18 2017
%Y Same expression with exponent b instead of 4: A001792 (b=1), A003583 (b=2), A007403 (b=3), A294435 (b=4), A294436 (b=5).
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 17 2017
|
