A277639 Double binomial partial sums of A007004.

1, 4, 43, 718, 14779, 344452, 8725093, 234594766, 6596287411, 192032529388, 5747827847545, 175986201591130, 5490952102178725, 174077883157001740, 5594651323154783515, 181946073109880839450, 5978730547304013537475, 198263347772478727193740, 6628299876919271425393105, 223211734849614639629628010, 7566093949269408444819804937

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Sum_{k=0..n} binomial(n,k)^2*multinomial(k,k,k)/(k+1).

a(n) = hypergeometric(1/3,2/3,-n,-n;1,1,2;27).

Double e.g.f.: BesselI(0,2*sqrt(t))*hypergeometric(1/3,2/3;1,1,2;27*t).

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

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

Mathematica

Table[Sum[Binomial[n, k]^2 Multinomial[k, k, k]/(k+1), {k, 0, n}], {n, 0, 100}]

Prog

(Maxima) makelist(sum(binomial(n, k)^2*multinomial_coeff(k, k, k)/(k+1), k, 0, n), n, 0, 12);

Crossrefs

Cf. A007004.

Sequence in context: A130545 A027311 A198205 * A326432 A074702 A197717

Adjacent sequences: A277636 A277637 A277638 * A277640 A277641 A277642

Keyword

nonn

Author

Emanuele Munarini, Oct 25 2016

Status

approved