A009014 -id:A009014 - cp's OEIS frontend

A009769 Expansion of tanh(log(1+1/x)).

1, 0, -4, 24, -96, 0, 5760, -80640, 645120, 0, -116121600, 2554675200, -30656102400, 0, 11158821273600, -334764638208000, 5356234211328000, 0, -3278015337332736000, 124564582818643968000, -2491291656372879360000, 0

Offset: 0

R. H. Hardin

Cf. A009014, A009116.

nn = 26; Range[0, nn]! CoefficientList[Series[Tanh[Log[1 + 1/x]], {x, 0, nn}], x] (* T. D. Noe, Oct 05 2011 *)

E.g.f.: (2*x+1)/(2*x^2+2*x+1) = 1-4*x^2/2!+24*x^3/3!-96*x^4/4!+....
Recurrence: a(n) = -2*n*a(n-1)-2*n*(n-1)*a(n-2), a(0) = 1, a(1) = 0.
a(n) = -n!/2*((-1+i)^(n+1) + (-1-i)^(n+1)) = -n!*sqrt(2)^(n+1)* cos(3*Pi*(n+1)/4).
a(n) = 2^n*A009014(n). a(n) = -n!*A009116(n+1).
For x > -1/2 we have (2*x+1)/(2*x^2+2*x+1) = 2*int {t = 0..inf} exp(-t*(2*x+1))*cos(t). Using this we obtain a(n) = 2*(-2)^n*int {t = 0..inf} t^n*exp(-t)*cos(t). - Peter Bala, Oct 05 2011

Extended with signs by Olivier Gérard, Mar 15 1997

A328286 Expansion of e.g.f. -log(1 - x - x^2/2).

Original entry on oeis.org

1, 2, 5, 21, 114, 780, 6390, 61110, 667800, 8210160, 112152600, 1685237400, 27624920400, 490572482400, 9381882510000, 192238348302000, 4201639474032000, 97572286427616000, 2399151995223984000, 62268748888378032000, 1701213856860117600000

Offset: 1

Ilya Gutkovskiy, Oct 11 2019

nonn

Cf. A009014, A039647, A080040, A080599 (exponential transform).

b:= proc(n) b(n):= n! * (<<1|1>, <1/2|0>>^n)[1, 1] end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-add(
      binomial(n, j)*j*b(n-j)*a(j), j=1..n-1)/n)
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Oct 11 2019

nmax = 21; CoefficientList[Series[-Log[1 - x - x^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest
FullSimplify[Table[(n - 1)! ((1 - Sqrt[3])^n + (1 + Sqrt[3])^n)/2^n, {n, 1, 21}]]

my(x='x+O('x^25)); Vec(serlaplace(-log(1 - x - x^2/2))) \\ Michel Marcus, Oct 11 2019

a(n) = (n - 1)! * ((1 - sqrt(3))^n + (1 + sqrt(3))^n) / 2^n.

D-finite with recurrence +2*a(n) +2*(-n+1)*a(n-1) -(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 20 2021

cp's OEIS Frontend

A009769 Expansion of tanh(log(1+1/x)).

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