A304788 Expansion of e.g.f. exp(Sum_{k>=1} binomial(2*k,k)*x^k/(k + 1)!).

1, 1, 3, 12, 59, 343, 2295, 17307, 144751, 1326377, 13189945, 141271298, 1619488645, 19766050827, 255693112641, 3492065507376, 50180426293255, 756444290843433, 11930511611596861, 196404976143077964, 3367697323914503113, 60029614473492823771, 1110430594720934758781

Offset: 0

Author

Ilya Gutkovskiy, May 18 2018

Keywords

• nonn

Comments

Exponential transform of A000108.

Examples

			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! + ...
		

Links

• N. J. A. Sloane, Transforms
• Eric Weisstein's World of Mathematics, Catalan Number

Crossrefs

Cf. A000108, A001517, A002212, A007317, A080893, A213507, A243953, A302197.

Programs

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}]

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).