A101683 Write exp(sqrt(1+x)-1) = Sum c(n) x^n/n!; then a(n) = numerator of c(n).
1, 1, 0, 1, -5, 9, -329, 3655, -11961, 721315, -12310199, 29326887, -4939227215, 113836841041, -356357531655, 77087063678521, -2238375706930349, 17366683494629835, -2294640596998068569, 80381887628910919255
Offset: 0
Examples
exp(sqrt(1+x)-1) = 1+(1/2)*x+(1/48)*x^3-(5/384)*x^4+(3/320)*x^5-(329/46080)*x^6+(731/129024)*x^7-(1329/286720)*x^8+... - From _N. J. A. Sloane_, Aug 29 2012
Links
- Robert Israel, Table of n, a(n) for n = 0..404
Crossrefs
Denominators are 2^A101684(n).
Programs
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Maple
c[0]:= 1: c[1]:= 1/2: for n from 2 to 100 do c[n]:= (c[n-2]-(4*n-6)*c[n-1])/4 od: seq(numer(c[n]),n=0..100); # Robert Israel, Nov 30 2023
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Mathematica
With[{nn=20},Numerator[CoefficientList[Series[Exp[Sqrt[1+x]-1],{x,0,nn}],x]Range[0,nn]!]] (* Harvey P. Dale, Aug 29 2012 *) -
PARI
my(x='x+O('x^30)); apply(numerator, Vec(serlaplace(exp(sqrt(1+x)-1)))) \\ Michel Marcus, Nov 30 2023
Formula
Numerator of c(n) satisfying c(n) = (c(n-2) - (4*n-6)*c(n-1))/4, c(0) = 1, c(1) = 1/2. - Robert Israel, Nov 30 2023
Extensions
Definition clarified by N. J. A. Sloane, Aug 29 2012
Comments