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%I #21 Aug 24 2022 09:01:03
%S 1,4,43,718,14779,344452,8725093,234594766,6596287411,192032529388,
%T 5747827847545,175986201591130,5490952102178725,174077883157001740,
%U 5594651323154783515,181946073109880839450,5978730547304013537475,198263347772478727193740,6628299876919271425393105,223211734849614639629628010,7566093949269408444819804937
%N Double binomial partial sums of A007004.
%F a(n) = Sum_{k=0..n} binomial(n,k)^2*multinomial(k,k,k)/(k+1).
%F a(n) = hypergeometric(1/3,2/3,-n,-n;1,1,2;27).
%F Double e.g.f.: BesselI(0,2*sqrt(t))*hypergeometric(1/3,2/3;1,1,2;27*t).
%F D-finite with recurrence: n^2*(n+1)^2*(1058*n^4 - 7061*n^3 + 16158*n^2 - 14048*n + 3284)*a(n) = 2*n*(30682*n^7 - 219052*n^6 + 555798*n^5 - 545060*n^4 + 16565*n^3 + 323730*n^2 - 206943*n + 39408)*a(n-1) - (834762*n^8 - 7954803*n^7 + 30596846*n^6 - 59518007*n^5 + 57023894*n^4 - 13636388*n^3 - 20674168*n^2 + 16952656*n - 3600432)*a(n-2) + 2*(n-2)^2*(744832*n^6 - 5313736*n^5 + 13458434*n^4 - 12947434*n^3 - 64535*n^2 + 6504872*n - 2110473)*a(n-3) - 676*(n-3)^2*(n-2)^2*(1058*n^4 - 2829*n^3 + 1323*n^2 + 1317*n - 609)*a(n-4). - _Vaclav Kotesovec_, Oct 30 2016
%F a(n) ~ sqrt(205/162 + 1939/(729*sqrt(3))) * (28+6*sqrt(3))^n / (Pi^(3/2)*n^(5/2)). - _Vaclav Kotesovec_, Oct 30 2016
%t Table[Sum[Binomial[n, k]^2 Multinomial[k,k,k]/(k+1), {k,0,n}], {n,0,100}]
%o (Maxima) makelist(sum(binomial(n,k)^2*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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