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A336271

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

1, 2, 10, 92, 1354, 29252, 873964, 34555880, 1748176714, 110183215988, 8467704986260, 779536758060920, 84699429189141100, 10725613123706081720, 1565870044943751242440, 261092436660169105108592, 49312362996510562406915914, 10473104312824253527997052500

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

FORMULA

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

a(n) ~ (n!)^2 * n / (BesselJ(1, 2*sqrt(r))^2 * r^(n+1)), where r = BesselJZero(0,1)^2 / 4 = A115368^2/4 = 1.4457964907366961302939989396139517587... - Vaclav Kotesovec, Jul 15 2020

MATHEMATICA

a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k]^2 Binomial[2 k, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

nmax = 17; CoefficientList[Series[1/BesselJ[0, 2 Sqrt[x]]^2, {x, 0, nmax}], x] Range[0, nmax]!^2

CROSSREFS

Cf. A000275, A000984, A002893.

Column k=2 of A340986.

Sequence in context: A111773 A289020 A195415 * A181084 A063385 A293709

Adjacent sequences: A336268 A336269 A336270 * A336272 A336273 A336274

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Jul 15 2020

STATUS

approved

URL: https://oeis.org/A336271

⇱ A336271 - OEIS