A214447 - OEIS

%I #13 Mar 19 2020 16:37:01

%S 1,-2,0,40,0,-4032,0,933504,0,-385848320,0,249576198144,0,

%T -232643283353600,0,295306112919306240,0,-489743069731226910720,0,

%U 1028154317960939805081600,0,-2665182817368374114506506240,0,8360422228704533182913131315200

%N (-2)^n * Euler_polynomial(n,1) * binomial(2*n,n).

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

%C Also a(n) = -A162660(2*n,n) for n > 0. - _Peter Luschny_, Jul 23 2012

%F 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.

%F 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

%p A214447 := n -> binomial(2*n,n)*(-2)^n*euler(n,1):

%p seq(A214447(n), n=0..23);

%o (Sage)

%o from mpmath import mp, fac2

%o mp.dps = 32

%o 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))

%o print([int(A214447(n)) for n in (0..19)]) # _Peter Luschny_, Jul 23 2012

%Y A081733, A214445.

%K sign

%O 0,2

%A _Peter Luschny_, Jul 18 2012