A009744 Expansion of e.g.f. tan(x)*sin(x) (even powers only).

0, 2, 4, 62, 1384, 50522, 2702764, 199360982, 19391512144, 2404879675442, 370371188237524, 69348874393137902, 15514534163557086904, 4087072509293123892362, 1252259641403629865468284, 441543893249023104553682822, 177519391579539289436664789664

Offset

0,2

Links

T. D. Noe, Table of n, a(n) for n = 0..100

Formula

G.f.: 1/G(0) - 1/(1+x) where G(k) = 1 - x*(2*k+1)^2/(1 - x*(2*k+2)^2/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 05 2013

G.f.: 1/G(0) - 1/(1+x) where G(k) = 1 - x*(k+1)^2/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 09 2013

a(n) ~ (2*n)! * 4^(n+1) / Pi^(2*n+1). - Vaclav Kotesovec, Jan 24 2015

Conjectural o.g.f.: Sum_{n >= 0} 4*x/2^n * Sum_{k = 0..n} (-1)^k*(k+1)*binomial(n,k)/( (1 + x*(2*k + 1)^2)*(1 + x*(2*k + 3)^2) ) = 2*x + 4*x^2 + 62*x^3 + 1384*x^4 + .... - Peter Bala, Mar 03 2015

From Peter Luschny, Jun 13 2021: (Start)

a(n) = (-1)^n*(Euler(2*n) - 1).

a(n) ~ 4^(2*n + 3/2)*exp(1/(24*n) - 2*n)*(n/Pi)^(2*n + 1/2). (End)

Maple

seq((2*i)!*coeff(series(tan(x)*sin(x), x, 30), x, 2*i), i=0..14); # Peter Luschny, Jul 14 2012

Mathematica

nn = 30; t = Range[0, nn]! CoefficientList[Series[Tan[x]*Sin[x], {x, 0, nn}], x]; Take[t, {1, nn, 2}] (* T. D. Noe, Jul 15 2012 *)

Prog

(Sage) # Variant of an algorithm of L. Seidel (1877) with a(0) = 1.
def A009744_list(n) :
    dim = 2*n; E = matrix(ZZ, dim); E[0, 0] = 1
    for m in (1..dim-1) :
        if m % 2 == 0 :
            E[m, 0] = 1;
            for k in range(m-1, -1, -1) :
                E[k, m-k] = E[k+1, m-k-1] - E[k, m-k-1]
        else :
            E[0, m] = 1;
            for k in range(1, m+1, 1) :
                E[k, m-k] = E[k-1, m-k+1] + E[k-1, m-k]
    return [(-1)^(k//2)*E[0, k] for k in range(dim) if is_even(k)]
A009744_list(14)  # Peter Luschny, Jul 14 2012
(PARI) x='x+O('x^50); v=Vec(serlaplace(tan(x)*sin(x))); concat([0], vector(#v\2, n, v[2*n-1])) \\ G. C. Greubel, Mar 04 2018

Crossrefs

Cf. A029582, A099023.

Sequence in context: A139160 A367287 A245164 * A124592 A275203 A232444

Adjacent sequences: A009741 A009742 A009743 * A009745 A009746 A009747

Keyword

nonn,easy

Author

R. H. Hardin

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997

Status

approved