A241242 a(n) = -2^(2*n+1)*(E(2*n+1, 1/2) + E(2*n+1, 1) + 2*(E(2*n+2, 1/2) + E(2*n+2, 1))), where E(n,x) are the Euler polynomials.

0, -3, 45, -1113, 42585, -2348973, 176992725, -17487754833, 2195014332465, -341282303124693, 64397376340013805, -14499110277050234553, 3840151029102915908745, -1182008039799685905580413, 418424709061213506712209285, -168805428822414120140493978273

Offset: 0

Author

Peter Luschny, Apr 17 2014

Keywords

• sign

Crossrefs

Cf. A000182, A000364, A028296, A240559, A147315, A186370.

Programs

Maple

A241242 := proc(n) e := n -> euler(n,1/2) + euler(n,1); -2^(2*n+1)*(e(2*n+1) + 2*e(2*n+2)) end: seq(A241242(n),n=0..15);

Mathematica

Array[-2^(2 # + 1)*(EulerE[2 # + 1, 1/2] + EulerE[2 # + 1, 1] + 2 (EulerE[2 # + 2, 1/2] + EulerE[2 # + 2, 1])) &, 16, 0] (* Michael De Vlieger, May 24 2018 *)

Formula

a(n) = A240559(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1).

a(n) = Sum_{k=0..2*n+1} (-1)^(2*n+1-k)*binomial(2*n+1, k)*2^k*(E(k, 1/2) + 2*E(k+1, 0)) where E(n,x) are the Euler polynomials.

a(n) = Sum_{k=0..2*n+1} (-1)^(2*n+1-k)*binomial(2*n+1, k)*(skp(k, 0) + skp(k+1, -1)), where skp(n, x) are the Swiss-Knife polynomials A153641.

a(n) = Bernoulli(2*n + 2) * 4^(n+1) * (1 - 4^(n+1)) / (2*n + 2) - EulerE(2*n + 2), where EulerE(2*n) is A028296. - Daniel Suteu, May 22 2018

a(n) = (-1)^(n+1) * (A000182(n+1) - A000364(n+1)). - Daniel Suteu, Jun 23 2018