A101684 Exponent of 2 in denominators of expansion of e.g.f.: exp(-1 + sqrt(1+x)).

0, 1, 0, 3, 4, 3, 6, 7, 6, 9, 10, 8, 12, 13, 11, 15, 16, 15, 18, 19, 18, 21, 22, 19, 24, 25, 22, 27, 28, 27, 30, 31, 30, 33, 34, 32, 36, 37, 35, 39, 40, 39, 42, 43, 42, 45, 46, 42, 48, 49, 45, 51, 52, 51, 54, 55, 54, 57, 58, 56, 60, 61, 59, 63, 64, 63, 66, 67, 66, 69, 70, 67

Offset: 0

Author

Ralf Stephan, Dec 13 2004

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

First differences are in A101685. Cf. A101683.

Programs

Maple

S:= series(exp(-1+sqrt(1+x)), x, 201):
seq(padic:-ordp(denom(n!*coeff(S,x,n)),2), n=0..200); # Robert Israel, Nov 30 2023

Mathematica

IntegerExponent[Denominator[With[{nn=80},CoefficientList[ Series[ Exp[ -1+Sqrt[1+x]],{x,0,nn}],x] Range[0,nn]!]],2] (* Harvey P. Dale, Aug 04 2021 *)

PARI

my(x='x+O('x^30)); apply(x->(valuation(denominator(x), 2)), Vec(serlaplace(exp(sqrt(1+x)-1)))) \\ Michel Marcus, Nov 30 2023

Formula

2^a(n) is denominator of c(n) where 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 Harvey P. Dale, Aug 04 2021