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%I #15 Oct 30 2016 05:42:25
%S 1,4,34,454,7384,133510,2583958,52468078,1104191608,23891534758,
%T 528527606698,11905777228618,272269758961018,6306419621308618,
%U 147677930682023818,3491114167267938298,83217317955857060788,1998209467779811473538,48293984598611551487038,1174012306200941229710038,28689784424223599507417938
%N Partial sums of A007004.
%F a(n) = Sum_{k=0..n} multinomial(k,k,k)/(k+1).
%F Recurrence: (n+2)*(n+3)*a(n+2)-(28*n^2+86*n+66)*a(n+1)+3*(3*n+5)*(3*n+4)*a(n)=0.
%F a(n) = hypergeometric(1/3,2/3;2;27)-(multinomial(n+1,n+1,n+1)/(n+2)) * hypergeometric[1,n+4/3,n+5/3;n+2,n+3;27).
%F a(n) = 0 (mod 2) and a(n) = 1 (mod 3), for all natural n.
%F G.f.: hypergeometric(1/3,2/3;2;27*t)/(1-t).
%F a(n) ~ 3^(3*n+7/2) / (52*Pi*n^2). - _Vaclav Kotesovec_, Oct 30 2016
%t Table[Sum[Multinomial[k, k, k]/(k + 1), {k, 0, n}], {n, 0, 100}]
%o (Maxima) makelist(sum(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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