A372513 Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(BesselJ(0,2*sqrt(2*x))) / 2.

0, 1, 2, 16, 264, 7296, 302720, 17587200, 1362399360, 135693537280, 16893684928512, 2570631845806080, 469393033744588800, 101294080603625226240, 25502237392032633323520, 7408331513180811911233536, 2459543337577081650719784960, 925435622656059412145504256000

Offset

0,3

Links

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

Formula

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

a(n) = 2^(n-1) * A002190(n).

Mathematica

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

Crossrefs

Cf. A002190.

Sequence in context: A304317 A351918 A326272 * A283685 A197458 A050974

Adjacent sequences: A372510 A372511 A372512 * A372515 A372516 A372517

Keyword

nonn

Author

Ilya Gutkovskiy, May 04 2024

Status

approved