A274246 - OEIS

%I #26 Dec 27 2022 02:29:45

%S 1,3,22,230,3048,48152,875536,17907024,405320320,10030449536,

%T 268836428544,7744939895552,238352004594688,7795463142466560,

%U 269761049981827072,9839883848966985728,377091995258812268544,15139047281589466136576,635088889901946682408960,27775758544209632635060224,1263876454164193257295446016

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

%H Robert Israel, <a href="/A274246/b274246.txt">Table of n, a(n) for n = 0..417</a>

%F 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).

%F 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))).

%F 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

%F a(n) = 2^n * Hypergeometric3F1([-n, -n, -n], [1], -1/2). - _G. C. Greubel_, Dec 27 2022

%p 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):

%p map(f, [$0..30]); # _Robert Israel_, Nov 16 2017

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

%o (Magma) [(&+[Binomial(n,j)^3*Factorial(j)*2^(n-j): j in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Dec 27 2022

%o (SageMath)

%o def A274246(n): return sum(binomial(n,j)^3*factorial(j)*2^(n-j) for j in range(n+1))

%o [A274246(n) for n in range(31)] # _G. C. Greubel_, Dec 27 2022

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

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Oct 12 2016