A214447 a(n) = (-2)^n * Euler_polynomial(n,1) * binomial(2*n,n).

1, -2, 0, 40, 0, -4032, 0, 933504, 0, -385848320, 0, 249576198144, 0, -232643283353600, 0, 295306112919306240, 0, -489743069731226910720, 0, 1028154317960939805081600, 0, -2665182817368374114506506240, 0, 8360422228704533182913131315200

Offset: 0

Author

Peter Luschny, Jul 18 2012

Keywords

• sign

Comments

Central column of the Euler tangent triangle, a(n) = A081733(2*n,n).

Also a(n) = -A162660(2*n,n) for n > 0. - Peter Luschny, Jul 23 2012

Crossrefs

Cf. A081733, A214445.

Programs

Maple

A214447 := n -> binomial(2*n,n)*(-2)^n*euler(n,1):
seq(A214447(n), n=0..23);

Sage

from mpmath import mp, fac2
mp.dps = 32
def A214447(n) : return (n+1)*2^n*fac2(2*n-1)*add(add((-1)^(k-j)*(k-j)^n / (factorial(j)*factorial(n-j+1)) for j in (0..k)) for k in (1..n))
print([int(A214447(n)) for n in (0..19)]) # Peter Luschny, Jul 23 2012

Formula

a(n) = [x^n] (skp(2*n,x+1)-skp(2*n,x-1))/2 where skp(n,x) are the Swiss-Knife polynomials A153641.

a(n) = (n+1)*2^n*(2*n-1)!!*Sum_{k=1..n} Sum_{j=0..k} (-1)^(k-j)*(k-j)^n/(j!*(n-j+1)!) for n > 0. - Peter Luschny, Jul 23 2012