A362161 -id:A362161 - cp's OEIS frontend

A362154 Expansion of 1/(1 + x * sqrt(1-4*x)).

1, -1, 3, -3, 11, -5, 51, 35, 335, 689, 3055, 9073, 33891, 114425, 413623, 1464661, 5304803, 19199859, 70161499, 257270227, 948639463, 3511150611, 13047799395, 48649126799, 181968324231, 682577156519, 2567180111683, 9678596662499, 36571577629911

Offset: 0

Seiichi Manyama, Apr 09 2023

Cf. A039834, A362155.

Cf. A000984, A362156, A362161.

my(N=30, x='x+O('x^N)); Vec(1/(1+x*sqrt(1-4*x)))

a(n) = (-1)^n * Sum_{k=0..n} 4^k * binomial((n-k)/2,k).

A362165 Expansion of e.g.f. exp(-x * sqrt(1-2*x)).

Original entry on oeis.org

1, -1, 3, -4, 25, 24, 721, 5942, 82209, 1186280, 19956241, 373942194, 7768988833, 177018731876, 4389959146665, 117700102748654, 3392361669670081, 104592876707106672, 3434908281762030049, 119702402508549928490, 4411764405039931048641

Offset: 0

Seiichi Manyama, Apr 10 2023

sign

Cf. A115329, A362161, A362163.

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*sqrt(1-2*x))))

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

cp's OEIS Frontend

A362154 Expansion of 1/(1 + x * sqrt(1-4*x)).

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