A117443 Expansion of e.g.f.: exp(x)/(cos(x) + sin(x)).

1, 0, 2, -4, 28, -160, 1272, -11184, 114448, -1309440, 16680992, -233587264, 3569157568, -59075960320, 1053056675712, -20111857791744, 409715696197888, -8868323731660800, 203247024658514432, -4916860703228314624, 125206830774036241408, -3347784042587048058880

Offset: 0

Author

Paul Barry, Mar 16 2006

Keywords

• easy
• sign

Comments

Row sums of number triangle A117442. Binomial transform of alternating sign Springer numbers (-1)^n*A001586(n).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..425

Crossrefs

Cf. A117442.

Programs

Mathematica

CoefficientList[Series[E^x/(Cos[x]+Sin[x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
A117442[n_, k_]:= (-1)^(n-k)*Binomial[n, k]*Abs[Numerator[EulerE[n-k, 1/4]]]; Table[Sum[A117442[n, k], {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jun 02 2021 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(x)/(cos(x) + sin(x)))) \\ Michel Marcus, Jun 02 2021

Sage

@CachedFunction
def f(n): return (-1/4)^n*sum( binomial(n, j)*2^j*euler_number(j) for j in (0..n) ) # f(n) = Euler(n, 1/4)
def A117443(n): return sum( (-1)^(n+k)*binomial(n,k)*abs(numerator(f(n-k))) for k in (0..n) )
[A117443(n) for n in (0..30)] # G. C. Greubel, Jun 02 2021

Formula

E.g.f.: 1/Q(0); Q(k)=1-(x^2)/((4*k+1)*(2*k+1)+2*x*(4*k+1)*(2*k+1)/(4*k+3-2*x-x*(4*k+3)/(x-(4*k+4)/Q(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 28 2011

G.f.: 1/Q(0) where Q(k) = 1 + 4*k*x - 2*x^2*(2*k + 1)^2/( 1 + (4*k+2)*x - 2*x^2*(2*k + 2)^2/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 10 2013

a(n) ~ (-1)^n * n! * 2^(2*n+3/2) / (Pi^(n+1) * exp(Pi/4)). - Vaclav Kotesovec, Aug 04 2014

a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * abs(numerator( Euler(n-k, 1/4) )), where Euler(n, x) is the Euler number polynomial. - G. C. Greubel, Jun 02 2021