A336608 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-x) / BesselJ(0,2*sqrt(x)).

1, 0, 1, 4, 51, 856, 21435, 725796, 32132499, 1800176176, 124511280723, 10420458131260, 1037868062069113, 121317006426807192, 16446390218708245393, 2559445829942874207804, 453188354421968867989395, 90587738500599611033753184

Offset

0,4

Links

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

Formula

a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000275(k) / k!.

a(n) ~ Pi * 2^(2*n+2) * n^(2*n+1) / (BesselJ(1, r) * exp(2*n + r^2/4) * r^(2*n+1)), where r = BesselJZero(0,1) = A115368. - Vaclav Kotesovec, Feb 09 2026

Mathematica

nmax = 17; CoefficientList[Series[Exp[-x]/BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
A000275[0] = 1; A000275[n_] := A000275[n] = -Sum[(-1)^(n - k) Binomial[n, k]^2 A000275[k], {k, 0, n - 1}]; a[n_] := n! Sum[(-1)^(n - k) Binomial[n, k] A000275[k]/k!, {k, 0, n}]; Table[a[n], {n, 0, 17}]

Crossrefs

Cf. A000275, A002720, A009940, A336606.

Cf. A115368.

Sequence in context: A347921 A328931 A343572 * A376105 A349653 A235325

Adjacent sequences: A336605 A336606 A336607 * A336609 A336610 A336611

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 27 2020

Status

approved