A240559 a(n) = -2^n*(E(n, 1/2) + E(n, 1) + (n mod 2)*2*(E(n+1, 1/2) + E(n+1, 1))), where E(n, x) are the Euler polynomials.

0, 0, 1, -3, -5, 45, 61, -1113, -1385, 42585, 50521, -2348973, -2702765, 176992725, 199360981, -17487754833, -19391512145, 2195014332465, 2404879675441, -341282303124693, -370371188237525, 64397376340013805, 69348874393137901, -14499110277050234553

Offset: 0

Author

Peter Luschny, Apr 17 2014

Keywords

• sign

Examples

			G.f. = x^2 - 3*x^3 - 5*x^4 + 45*x^5 + 61*x^6 - 1113*x^7 - 1385*x^8 + ...
		

Links

• L. Seidel, Über eine einfache Entstehungsweise der Bernoullischen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, Vol. 7 (1877), 157-187.

Crossrefs

Cf. A000364, A147315, A186370, A241242.

Cf. A239005, A239322.

Programs

Maple

A240559 := proc(n) euler(n,1/2) + euler(n,1); if n mod 2 = 1 then % + 2*(euler(n+1,1/2)+euler(n+1,1)) fi; -2^n*% end: seq(A240559(n),n=0..19);

Mathematica

skp[n_, x_] := Sum[Binomial[n, k]*EulerE[k]*x^(n-k), {k, 0, n}]; skp[n_, x0_?NumericQ] := skp[n, x] /. x -> x0; a[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*(skp[k, 0] + skp[k+1, -1]), {k, 0, n}]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 09 2014, after Peter Luschny *)

Sage

# Efficient computation with L. Seidel's boustrophedon transformation.
def A240559_list(n) :
    A = [0]*(n+1); A[0] = 1; R = [0]
    k = 0; e = 1; x = -1; s = -1
    for i in (0..n):
        Am = 0; A[k + e] = 0; e = -e;
        for j in (0..i): Am += A[k]; A[k] = Am; k += e
        if e == 1: x += 1; s = -s
        v = -A[-x] if e == 1 else A[-x] - A[x]
        if i > 1: R.append(s*v)
    return R
A240559_list(24)

Formula

a(2*n) = (-1)^(n+1)*A147315(2*n,1) = (-1)^(n+1)*A186370(2*n,2*n) =(-1)^(n+1)*A000364(n) for n>0.

a(2*n+1) = (-1)^n*A147315(2*n+1,2) = (-1)^n*A186370(2*n,2*n-1) = A241242(n).

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

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

a(n) = A239322(n) + A239005(n+1) - A239005(n). - Paul Curtz, Apr 18 2014

E.g.f.: 1 - sech(x) - tanh(x) + sinh(x)*sech(x)^2 = ((exp(-x)-1)*sech(x))^2 / 2. - Sergei N. Gladkovskii, Nov 20 2014

E.g.f.: (1 - sech(x)) * (1 - tanh(x)). - Michael Somos, Nov 22 2014