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
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 0..417
Programs
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Magma
[(&+[Binomial(n,j)^3*Factorial(j)*2^(n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Dec 27 2022
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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}] -
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
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