A362165 Expansion of e.g.f. exp(-x * sqrt(1-2*x)).
1, -1, 3, -4, 25, 24, 721, 5942, 82209, 1186280, 19956241, 373942194, 7768988833, 177018731876, 4389959146665, 117700102748654, 3392361669670081, 104592876707106672, 3434908281762030049, 119702402508549928490, 4411764405039931048641
Offset: 0
Keywords
Programs
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Maple
A362165 := proc(n) (-1)^n*n!*add(2^k * binomial((n-k)/2,k)/(n-k)!,k=0..n) ; end proc: seq(A362165(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*sqrt(1-2*x))))
Formula
a(n) = (-1)^n * n! * Sum_{k=0..n} 2^k * binomial((n-k)/2,k)/(n-k)!.
D-finite with recurrence a(n) +2*(-n+3)*a(n-1) +2*(-3*n+10)*a(n-2) +6*(n-2)*a(n-3) -9*(n-3)^2*a(n-4) -27*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Dec 04 2023