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

%I A214447 #18 Feb 19 2025 02:45:00
%S A214447 1,-2,0,40,0,-4032,0,933504,0,-385848320,0,249576198144,0,
%T A214447 -232643283353600,0,295306112919306240,0,-489743069731226910720,0,
%U A214447 1028154317960939805081600,0,-2665182817368374114506506240,0,8360422228704533182913131315200
%N A214447 a(n) = (-2)^n * Euler_polynomial(n,1) * binomial(2*n,n).
%C A214447 Central column of the Euler tangent triangle, a(n) = A081733(2*n,n).
%C A214447 Also a(n) = -A162660(2*n,n) for n > 0. - _Peter Luschny_, Jul 23 2012
%F A214447 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 A214447 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 A214447 := n -> binomial(2*n,n)*(-2)^n*euler(n,1):
%p A214447 seq(A214447(n), n=0..23);
%o A214447 (Sage)
%o A214447 from mpmath import mp, fac2
%o A214447 mp.dps = 32
%o A214447 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 A214447 print([int(A214447(n)) for n in (0..19)]) # _Peter Luschny_, Jul 23 2012
%Y A214447 Cf. A081733, A214445.
%K A214447 sign
%O A214447 0,2
%A A214447 _Peter Luschny_, Jul 18 2012