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%I
%S 1,4,37,520,8803,165292,3320023,69943804,1526981575,34271322316,
%T 786371041603,18372739163632,435772652437381,10468369504009060,
%U 254238148448959729,6233226769739934964,154092763036678601551,3837301178450916902428
%N Binomial partial sums of sequence A007004.
%F a(n) = Sum_{k=0..n} binomial(n,k) * multinomial(k,k,k)/(k+1).
%F a(n) = hypergeometric(1/3,2/3,-n; 1,2; -27).
%F a(n) == 1 (mod 3) for all natural n.
%F E.g.f.: exp(t) * hypergeometric(1/3,2/3; 1,2; 27*t).
%F From _Vaclav Kotesovec_, Oct 26 2016: (Start)
%F Recurrence: n*(n+1)*a(n) = 2*(3*n-1)*(5*n-3)*a(n-1) - (n-1)*(57*n-56)*a(n-2) + 28*(n-2)*(n-1)*a(n-3).
%F a(n) ~ 2^(2*n+3) * 7^(n+2) / (3^(11/2) * Pi * n^2).
%F (End)
%F Diff. eq. satisfied by the ordinary g.f.: t*(1-t)^2*(1-28*t)*A''(t)+2*(1-t)*(1-2*t)*(1-28*t)*A'(t)-2*(4-29*t+28*t^2)*A(t)=0. - _Emanuele Munarini_, Oct 28 2016
%t Table[Sum[Binomial[n, k] Multinomial[k, k, k]/(k+1), {k, 0, n}], {n, 0, 100}]
%o (Maxima) makelist(sum(binomial(n, k)*multinomial_coeff(k, k, k)/(k+1), k, 0, n), n, 0, 12);
%Y Cf. A007004.
%K nonn
%O 0,2
%A _Emanuele Munarini_, Oct 25 2016
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