A274246 a(n) = Sum_{k=0..n} binomial(n, k)^3 * 2^(n-k) * k!.

1, 3, 22, 230, 3048, 48152, 875536, 17907024, 405320320, 10030449536, 268836428544, 7744939895552, 238352004594688, 7795463142466560, 269761049981827072, 9839883848966985728, 377091995258812268544, 15139047281589466136576, 635088889901946682408960, 27775758544209632635060224, 1263876454164193257295446016

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..417

Formula

Recurrence: n*(2*n - 5)*a(n) = (6*n^3 - 13*n^2 - 8*n + 6)*a(n-1) - (n-1)*(6*n^3 - 51*n^2 + 124*n - 90)*a(n-2) + (n-2)^3*(n-1)*(2*n - 3)*a(n-3).

a(n) ~ n^(n - 1/6) * exp(3*2^(1/3)*n^(2/3) - 2^(2/3)*n^(1/3) - n + 2/3) / (2^(5/6)*sqrt(3*Pi)) * (1 + 31*2^(1/3)/(27*n^(1/3)) + 3437/(3645*2^(1/3) * n^(2/3))).

Sum_{n>=0} a(n) * x^n / n!^3 = BesselI(0,2*sqrt(x)) * Sum_{n>=0} 2^n * x^n / n!^3. - Ilya Gutkovskiy, Jun 19 2022

a(n) = 2^n * Hypergeometric3F1([-n, -n, -n], [1], -1/2). - G. C. Greubel, Dec 27 2022

Maple

f:= gfun:-rectoproc({n*(2*n - 5)*a(n) = (6*n^3 - 13*n^2 - 8*n + 6)*a(n-1) - (n-1)*(6*n^3 - 51*n^2 + 124*n - 90)*a(n-2) + (n-2)^3*(n-1)*(2*n - 3)*a(n-3), a(0)=1, a(1)=3, a(2)=22}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Nov 16 2017

Mathematica

Table[Sum[Binomial[n, k]^3 * 2^(n-k) * k!, {k, 0, n}], {n, 0, 20}]

Prog

(Magma) [(&+[Binomial(n, j)^3*Factorial(j)*2^(n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Dec 27 2022
(SageMath)
def A274246(n): return sum(binomial(n, j)^3*factorial(j)*2^(n-j) for j in range(n+1))
[A274246(n) for n in range(31)] # G. C. Greubel, Dec 27 2022

Crossrefs

Cf. A000172, A087912, A216831, A241247, A277386.

Sequence in context: A073530 A120667 A196958 * A161567 A213109 A141006

Adjacent sequences: A274243 A274244 A274245 * A274247 A274248 A274249

Keyword

nonn

Author

Vaclav Kotesovec, Oct 12 2016

Status

approved