A352470 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1)^2 * a(n-2*k-1).

1, 1, 4, 37, 608, 15601, 576472, 28993693, 1904637184, 158352856129, 16253786050904, 2018684970206653, 298373110433984192, 51757706826973479697, 10412613242348421164400, 2404755328388872932588037, 631887117002962512609921024, 187441600433239155105076467457

Offset

0,3

Links

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

Formula

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

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

Mathematica

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

Crossrefs

Cf. A006154, A102221, A346220, A352467, A352471.

Sequence in context: A133471 A249607 A097998 * A177385 A155926 A094408

Adjacent sequences: A352467 A352468 A352469 * A352471 A352472 A352473

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 17 2022

Status

approved