A337825 a(0) = 0; a(n) = n^3 - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k)^3 * k * a(k).

0, 1, 6, -33, -512, 19405, 181116, -45817541, 771776384, 280415588121, -23151651942500, -3217963989270569, 816268626535923936, 38087192839910816485, -43268389662374707851552, 2822720920753640236252875, 3297662826737476255127428096, -833876355494162903256716734927

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + x * (BesselI(0,2*sqrt(x)) + sqrt(x) * BesselI(1,2*sqrt(x)))).

Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n^3 * x^n / (n!)^2).

Mathematica

a[0] = 0; a[n_] := a[n] = n^3 - (1/n) * Sum[Binomial[n, k]^2 (n - k)^3 k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Log[1 + x (BesselI[0, 2 Sqrt[x]] + Sqrt[x] BesselI[1, 2 Sqrt[x]])], {x, 0, nmax}], x] Range[0, nmax]!^2

Crossrefs

Cf. A002190, A101981, A300452, A337590, A337824, A337826.

Sequence in context: A121376 A354890 A046707 * A183466 A112289 A354479

Adjacent sequences: A337822 A337823 A337824 * A337826 A337827 A337828

Keyword

sign

Author

Ilya Gutkovskiy, Sep 24 2020

Status

approved