A189392 E.g.f. tan(x)/(1-tan(x)-tan(x)^2).

0, 1, 2, 14, 88, 856, 8912, 115184, 1644928, 26916736, 484527872, 9646440704, 208891091968, 4908327894016, 124094242629632, 3363100573005824, 97195058375262208, 2984961531977629696, 97056253813754888192, 3331252527903082348544

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..65 (corrected by Ray Chandler, Jan 19 2019)

Formula

E.g.f.: A(tan(x)) where A(x) = x/(1-x-x^2) is the g.f. of A000045.

a(n)=sum(k=1..n, (sum(i=ceiling((k-1)/2)..k-1, binomial(i,k-i-1)))*((-1)^(n-k)+1)*sum(j=k..n, binomial(j-1,k-1)*j!*2^(n-j-1)*(-1)^((n+k)/2+j)*stirling2(n,j)));

From Peter Bala, Mar 12 2013: (Start)

With an offset of -1 the e.g.f is 1/(cos(2*x) - 1/2*sin(2*x))^2 = 1 + 2*x + 14*x^2/2! + 88*x^3/3! + .... This relates this sequence to A185896. Define a sequence of polynomials R(n,x) by the recurrence R(n+1,x) = d/dx{(x^2 + 4)*R(n,x)} with R(1,x) = 1. Then a(n) = R(n,1). Let M be the array with the sequence [2,3,4,...] on the main superdiagonal and the sequence [4,8,12,...] on the main subdiagonal and 0's everywhere else. Then a(n+1) equals the sum of the entries in row 1 of M^n.

(End)

a(n) ~ n! / (5*(arctan(sqrt(5)/2-1/2))^(n+1)). - Vaclav Kotesovec, Jun 02 2013

G.f.: x/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - 5*x^2*(2*k+1)*(2*k+2)/( 1 - 2*x*(2*k+2) - 5*x^2*(2*k+2)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 23 2013

E.g.f.: W(0) -1, where W(k) = 1 + x/( 4*k+1 - x/( 1 - 4*x/( 4*k+3 + 4*x/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 27 2014

Example

G.f. = x + 2*x^2 + 14*x^3 + 88*x^4 + 856*x^5 + 8912*x^6 + 115184*x^7 + ...

Mathematica

nn = 20; Range[0, nn]! CoefficientList[Series[Tan[x]/(1 - Tan[x] - Tan[x]^2), {x, 0, nn}], x] (* T. D. Noe, Mar 14 2013 *)

Prog

(Maxima)
a(n):=sum((sum(binomial(i, k-i-1), i, ceiling((k-1)/2), k-1))*((-1)^(n-k)+1)*sum(binomial(j-1, k-1)*j!*2^(n-j-1)*(-1)^((n+k)/2+j)*stirling2(n, j), j, k, n), k, 1, n);
(PARI)  x='x+O('x^66); /* that many terms */
egf=tan(x)/(1-tan(x)-tan(x)^2); /* = x + x^2 + 7/3*x^3 + 11/3*x^4 +... */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 22 2011 */

Crossrefs

Cf. A185896.

Sequence in context: A065355 A162478 A348615 * A235374 A065892 A139183

Adjacent sequences: A189389 A189390 A189391 * A189393 A189394 A189395

Keyword

nonn

Author

Vladimir Kruchinin, Apr 21 2011

Status

approved