OFFSET
0,3
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..100
Peter Bala, The method of Graves for computing inverse functions.
Vladimir V. Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
FORMULA
a(n) = b(2*n-1), b(n) = Sum_{k = 1..n} ((-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))*((-1)^((k+3)/2)-(-1)^((3*k+3)/2))/(2*k!). - Vladimir Kruchinin, Apr 23 2011
a(n) = Sum_{m = 0..n} (Sum_{j = 0..2*n-2*m} binomial(j+2*m,2*m)*(j+2*m+1)!*2^(2*n-2*m-j)*(-1)^(n+j)*Stirling2(2*n+1,j+2*m+1))/(2*m+1)!. - Vladimir Kruchinin, Jan 21 2012
For n >= 1, set d(n, x) = (1 + x^2)*d/dx(d(n-1, x)) with d(0, x) = sin(x). Then a(n) = d(2*n-1, 0). - Peter Bala, Jan 24 2026
MAPLE
d := proc(n, x) option remember; if n = 0 then sin(x) else simplify( (1 + x^2)*diff( d(n-1, x), x) ) end if end proc:
seq( eval(d(2*n-1, x), x = 0), n = 1..20 ); # Peter Bala, Jan 24 2026
MATHEMATICA
With[{nn=50}, Take[CoefficientList[Series[Sin[Tan[x]], {x, 0, nn}], x] Range[ 0, nn-1]!, {2, -1, 2}]] (* Harvey P. Dale, Jul 25 2012 *)
PROG
(Maxima)
a(n):=b(2*n-1);
b(n):=sum(((-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)*((-1)^((k+3)/2)-(-1)^((3*k+3)/2))/(2*k!), k, 1, n); /* Vladimir Kruchinin, Apr 23 2011 */
a(n):=sum(sum(binomial(j+2*m, 2*m)*(j+2*m+1)!*2^(2*n-2*m-j)*(-1)^(n+j)*stirling2(2*n+1, j+2*m+1), j, 0, 2*n-2*m)/((2*m+1)!), m, 0, n); /* Vladimir Kruchinin, Jan 21 2012 */
CROSSREFS
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
Corrected name, Joerg Arndt, Apr 23 2011
More terms from Harvey P. Dale, Jul 25 2012
STATUS
approved
