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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).
5
0, 1, 4, 34, 576, 16296, 691408, 41069568, 3252707328, 331218217600, 42159307194624, 6558777387076608, 1224428872399488000, 270143735036619436032, 69534931015726331203584, 20651854796028308275851264, 7009822878720340562163007488, 2696576146784893519040303235072, 1166999997199470676471689819258880
OFFSET
0,3
LINKS
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 04 2020
STATUS
approved