A333981 a(0) = 0; a(n) = 2^(n-1) + (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * 2^(k-1) * (n-k) * a(n-k).

0, 1, 4, 34, 576, 16296, 691408, 41069568, 3252707328, 331218217600, 42159307194624, 6558777387076608, 1224428872399488000, 270143735036619436032, 69534931015726331203584, 20651854796028308275851264, 7009822878720340562163007488, 2696576146784893519040303235072, 1166999997199470676471689819258880

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..245

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = -log((3 - BesselI(0,2*sqrt(2*x))) / 2).

Mathematica

a[0] = 0; a[n_] := a[n] = 2^(n - 1) + (1/n) Sum[Binomial[n, k]^2 2^(k - 1) (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[-Log[(3 - BesselI[0, 2 Sqrt[2 x]])/2], {x, 0, nmax}], x] Range[0, nmax]!^2

Prog

(SageMath)
@CachedFunction
def a(n): return 0 if (n==0) else 2^(n-1) + (1/n)*sum(binomial(n, k)^2 *2^(k-1)*(n-k)*a(n-k) for k in (1..n-1)) # a= A333981
[a(n) for n in (0..30)] # G. C. Greubel, Jun 09 2022

Crossrefs

Cf. A102223, A123227, A333982, A333983, A333984, A333985, A337592.

Sequence in context: A081972 A158961 A134354 * A030243 A222789 A326206

Adjacent sequences: A333978 A333979 A333980 * A333982 A333983 A333984

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 04 2020

Status

approved