A003720 Expansion of e.g.f. tan(tan(tan(x))).

1, 6, 168, 10672, 1198080, 208521728, 51874413568, 17449541107712, 7622674735988736, 4193561606973095936, 2836052065377836597248, 2312174256451088534208512, 2236165580390456719589769216, 2530976708469616321520834969600, 3314110602212685014002135203840000

Offset: 0

Author

R. H. Hardin

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Programs

Mathematica

Rest@ Union[ Range[0, 25]! CoefficientList[ Series[Tan@ Tan@ Tan@ x, {x, 0, 25}], x]] (* Robert G. Wilson v *)

Maxima

a(n):=b(2*n+1); b(n):=sum((((-1)^(m-1)+1)*(sum(j!*2^(m-j-1)* (-1)^((m+1)/2+j) *stirling2(m,j), j,1,m))*sum((((-1)^(k-m)+1)*(sum(binomial(j-1,m-1)* j!*2^(k-j-1)*stirling2(k,j)*(-1)^((m+k)/2+j), j,m,k))*((-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!,k,m,n))/m!, m,1,n); /* Vladimir Kruchinin, Apr 23 2011 */

PARI

x='x+O('x^66); /* that many terms */
serlaplace(tan(tan(tan(x)))) /* show terms */ /* Joerg Arndt, Apr 26 2011 */

Formula

a(n) = b(2*n+1) where b(n) = sum(m=1..n, (((-1)^(m-1)+1)*(sum(j=1..m, j! *2^(m-j-1)*(-1)^((m+1)/2+j)*S2(m,j)))*sum(k=m..n,(((-1)^(k-m)+1)*(sum(j=m..k, C(j-1,m-1)*j!*2^(k-j-1)*S2(k,j)*(-1)^((m+k)/2+j)))*((-1)^(n-k)+1)* sum(j=k,n, C(j-1,k-1)*j!*2^(n-j-1)* (-1)^((n+k)/2+j)* S2(n,j)))/k!))/m!). - Vladimir Kruchinin, Apr 23 2011

a(n) ~ 8*(2*n+1)!/((4+Pi^2) * (1+arctan(Pi/2)^2) * (arctan(arctan(Pi/2)))^(2*n+2)). - Vaclav Kotesovec, Feb 16 2015

Extensions

Extended and formatted Mar 15 1997 by Olivier Gérard

Corrected definition, Joerg Arndt, Apr 26 2011

a(13)-a(14) from Alois P. Heinz, May 13 2012