A336639 Sum_{n>=0} a(n) * x^n / (n!)^2 = 1 / BesselJ(0,2*sqrt(x))^4.

1, 4, 36, 544, 12196, 377904, 15438816, 803602944, 51908768676, 4074743122384, 382079412133936, 42184889139337344, 5417567866536188896, 800808722921088352384, 135006904500993157933056, 25751088570167886107910144, 5517695042810314282550432676

Offset: 0

Author

Ilya Gutkovskiy, Jul 28 2020

Keywords

• nonn

Comments

In general, if k>=1 and Sum_{n>=0} a(n) * x^n / n!^2 = 1 / BesselJ(0, 2*sqrt(x))^k, then a(n) ~ n!^2 * n^(k-1) / ((k-1)! * r^(n + k/2) * BesselJ(1, 2*sqrt(r))^k), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4. - Vaclav Kotesovec, Jul 11 2025

Crossrefs

Cf. A000275, A002895, A336271, A336638.

Column k=4 of A340986.

Programs

Mathematica

nmax = 16; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^4, {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] HypergeometricPFQ[{1/2, -k, -k, -k}, {1, 1, 1/2 - k}, 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k)^2 * A002895(k) * a(n-k).

a(n) ~ n!^2 * n^3 / (6 * r^(n+2) * BesselJ(1, 2*sqrt(r))^4), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 11 2025