A242233 2^n*(C_n)^(1/2) in the Cauchy type product where C_n is the n-th Catalan number.

1, 1, 3, 11, 41, 137, 347, 611, 5777, 98321, 677363, -4192197, -134908871, -617972327, 22749265099, 449951818387, -632325203423, -163681108703199, -2324079456844573, 33233931805782635, 1734259111955765577, 14135975420529458857, -777499293367428199109

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

a(n) = 2^n*n!*[x^n](sqrt(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)))), where [x^n](f(x)) the coefficient of x^n in f(x).

For n > 0, a(n) = Sum_{k=1..n} a(n-k)*binomial(n,k)*(2*k)!*(3*k/(2*n)-1)*2^k/(k!*(k+1)!). - Tani Akinari, Nov 05 2024

Maple

f := sqrt(exp(2*x)*(BesselI(0, 2*x)-BesselI(1, 2*x)));
seq(2^n*n!*coeff(series(f, x, n+1), x, n), n=0..22);
# Second program with function g from A241885:
catalan := n -> binomial(2*n, n)/(n+1);
a := n -> 2^n*g(catalan, n); seq(a(n), n=0..22);

Mathematica

g[n_] := g[n] = (CatalanNumber[n] - Sum[Binomial[n, m] g[m] g[n - m], {m, 1, n - 1}])/2;
a[0] = 1; a[n_] := 2^n g[n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Aug 02 2019, from 2nd Maple program *)

Prog

(Maxima) a[n]:=if n=0 then 1 else sum(a[n-k]*binomial(n, k)*(2*k)!*(3*k/(2*n)-1)*2^k/(k!*(k+1)!), k, 1, n); makelist(a[n], n, 0, 50); /* Tani Akinari, Nov 05 2024 */

Crossrefs

Cf. A000108, A126156, A241885.

Sequence in context: A351428 A181863 A075276 * A294504 A086972 A218348

Adjacent sequences: A242230 A242231 A242232 * A242234 A242235 A242236

Keyword

sign

Author

Peter Luschny, May 08 2014

Status

approved