STATUS
proposed
approved
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Revision History for A242233
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2^n*(C_n)^(1/2) in the Cauchy type product where C_n is the n-th Catalan number.
(history; published version)
(history; published version)
STATUS
editing
proposed
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 *)
STATUS
approved
editing
STATUS
editing
approved
STATUS
editing
proposed
STATUS
editing
proposed
NAME
allocated for Peter Luschny2^n*(C_n)^(1/2) in the Cauchy type product where C_n is the n-th Catalan number.
DATA
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
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).
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 A126156:
catalan := n -> binomial(2*n, n)/(n+1);
a := n -> 2^n*g(catalan, n); seq(a(n), n=0..22);
KEYWORD
allocated
sign
AUTHOR
Peter Luschny, May 08 2014
STATUS
approved
editing
NAME
allocated for Peter Luschny
KEYWORD
allocated
STATUS
approved