A009562 Expansion of e.g.f. sin(x/cos(x)) (odd powers only).

1, 2, -4, -624, -37808, -2529440, -197245632, -17275207936, -1517532372736, -78275049887232, 24598365340871680, 16151599386896207872, 7394065277787966590976, 3266488097334689885904896, 1491592302825720723705577472, 718708462133271426457747783680

Offset: 1

Author

R. H. Hardin

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 1..240

Programs

Mathematica

With[{nmax = 50}, CoefficientList[Series[Sin[x/Cos[x]], {x, 0, nmax}], x]*Range[0, nmax - 1]!][[2 ;; ;; 2]] (* G. C. Greubel, Jan 21 2018 *)

Maxima

a(n):=2*(sum(binomial(2*n+1, 2*m+1)*sum(binomial(m+j-1/2, j)* 4^(n-m-j)*sum((i-j)^(2*n-2*m)*binomial(2*j, i)*(-1)^(n+j-i), i, 0, j), j, 0, n-m), m, 0, n-1))+(-1)^(n); /* Vladimir Kruchinin, Jun 29 2011 */

PARI

my(v=sin(x/cos(x))); forstep(n=1,default(seriesprecision),2,print1(polcoeff(v,n)*n!", ")) \\ Charles R Greathouse IV, Jun 27 2011

Formula

a(n+1) = (-1)^n + 2*Sum_{m=0..n-1} binomial(2*n+1, 2*m+1)*Sum_{j=0..n-m} binomial(m+j-1/2, j)*4^(n-m-j)*Sum_{i=0..j} (i-j)^(2*n-2*m)*binomial(2*j, i)*(-1)^(n+j-i). - Vladimir Kruchinin, Jun 29 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

a(15)-a(16) from G. C. Greubel, Jan 21 2018