OFFSET
0,3
COMMENTS
Exponential transform of A000108.
LINKS
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Catalan Number
FORMULA
E.g.f.: exp(Sum_{k>=1} A000108(k)*x^k/k!).
E.g.f.: exp(exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)) - 1).
EXAMPLE
E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 12*x^3/3! + 59*x^4/4! + 343*x^5/5! + 2295*x^6/6! + 17307*x^7/7! + ...
MAPLE
a:=series(exp(add(binomial(2*k, k)*x^k/(k+1)!, k=1..100)), x=0, 23): seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 26 2019
MATHEMATICA
nmax = 22; CoefficientList[Series[Exp[Sum[CatalanNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Exp[2 x] (BesselI[0, 2 x] - BesselI[1, 2 x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[CatalanNumber[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 18 2018
STATUS
approved