cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-15 of 15 results.

A060056 Nonzero numbers in expansion of ((tan(x))^4)/4! in (x^n)/n!.

Original entry on oeis.org

1, 40, 2016, 135680, 11977856, 1351633920, 190346960896, 32769353973760, 6776471542235136, 1658320063181619200, 474140484461265944576, 156647023437347542794240, 59229231136268698009993216, 25414555567107834838389882880, 12283447443202253774326182445056
Offset: 0

Views

Author

Wolfdieter Lang, Mar 16 2001

Keywords

Crossrefs

A059419 (fourth column without zeros), A059420.

Programs

  • PARI
    a059419(n, k) = n!*polcoef(tan(x+x*O(x^n))^k/k!, n);
a(n) = a059419(2*n+4, 4); \\ Seiichi Manyama, May 11 2025

Formula

a(n) = A059419(4+2*n,4).

A009737 Expansion of e.g.f. tan(x)*exp(tan(x)).

Original entry on oeis.org

0, 1, 2, 5, 20, 81, 438, 2477, 16680, 120481, 973034, 8496245, 80252732, 817734321, 8859646110, 102873611549, 1258403748432, 16372688411713, 223202277906386, 3213260867586149, 48295209177888356, 761792907575450385, 12510350648500199814, 214507625428065409805
Offset: 0

Views

Author

R. H. Hardin

Keywords

Links

Crossrefs

Cf. A052852, A059419, A008277.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
[0] cat Coefficients(R!( Laplace( Tan(x)*Exp(Tan(x)) ) )); // G. C. Greubel, Mar 09 2021
  • Maple
    m:= 30; S:= series(tan(x)*exp(tan(x)), x, m+1); seq(j!*coeff(S, x, j), j = 0..m); # G. C. Greubel, Mar 09 2021
  • Mathematica
    With[{nn=20},CoefficientList[Series[Tan[x]Exp[Tan[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 30 2011 *)
  • Maxima
    a(n):=sum((1+(-1)^(n-k))*sum(j!*stirling2(n,j)*2^(n-j-1)*(-1)^((n+k)/2+j)*binomial(j-1,k-1),j,k,n)/(k-1)!,k,1,n); /* Vladimir Kruchinin, Apr 19 2011 */
  • Sage
    [factorial(n)*( tan(x)*exp(tan(x)) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 09 2021

Formula

a(n) = Sum_{k=1..n} ((1+(-1)^(n-k))/(k-1)!) * Sum_{j=k..n} j! * Stirling2(n,j) * 2^(n-j-1)*(-1)^((n+k)/2+j)*binomial(j-1,k-1). - Vladimir Kruchinin, Apr 19 2011
a(n) = D^n(x*exp(x)) evaluated at x = 0, where D is the operator (1+x^2)*d/dx. Cf. A052852. a(n) = Sum_{k=1..n} k*A059419(n,k). - Peter Bala, Nov 25 2011

Extensions

Extended and signs tested by Olivier Gérard, Mar 15 1997

A002303 Generalized tangent numbers.

Original entry on oeis.org

16, 272, 3968, 56320, 814080, 12207360, 191431680, 3149752320, 54428774400, 987559372800, 18797300121600, 374883257548800, 7822865085235200, 170560590520320000, 3879770715684864000, 91945674412720128000
Offset: 4

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Letterio Toscano, Sulla Derivata di Ordinen della Funzione tg(x), Tohoku Math. J., 42 (1936), 144-154.

Crossrefs

Cf. A059419.

Programs

  • PARI
    a(n, k)=if(k<0,0,if(n==1 && k==1,1,if(k>n,0,(k-1)*a(n-1,k-1)+(k+1)*a(n-1,k+1))))
for(n=0,25,print1(a(n+6, n)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006

Formula

Ignoring the initial term a(4) = 16 and working with an offset of 0 the e.g.f. appears to be the rational function 16*(17+78*t+45*t^2)/(1-t)^10 = 272 + 3968*t + 56320*t^2/2! + ... . - Peter Bala, Apr 23 2012
This rational function occurs in the series reversion (x-t*tan(x))^(-1) = x/(1-t) + 2*t/(1-t)^4*x^3/3! + 8*t*(2+3*t)/(1-t)^7*x^5/5! + 16*t*(17+78*t+45*t^2)/(1-t)^10*x^7/7! + ..., which is the e.g.f. for the triangle A059419 read by diagonals. - Peter Bala, Apr 23 2012

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Nov 20 2006

A008308 Triangle of tangent numbers.

Original entry on oeis.org

1, 1, 2, 1, 8, 1, 16, 20, 1, 136, 40, 1, 272, 616, 70, 1, 3968, 2016, 112, 1, 7936, 28160, 5376, 168, 1, 176896, 135680, 12432, 240, 1, 353792, 1805056, 508640, 25872, 330, 1, 11184128, 11977856, 1595264, 49632, 440, 1, 22368256, 154918400, 59835776
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			Triangle begins:
    1;
    1;
    2,  1;
    8,  1;
   16, 20, 1;
  136, 40, 1;
  ...

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.

Crossrefs

Essentially the same triangle as A059419, which is the main entry for this triangle.
Row sums give A006229.

Programs

  • Mathematica
    T[n_, n_] = 1; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n - 1, k - 1] + k*(k + 1)*T[n - 1, k + 1]; T[, ] = 0;
row[n_] := DeleteCases[Table[T[n, k], {k, 1, n}] , 0];
Array[row, 13] // Flatten (* Jean-François Alcover, Nov 09 2017 *)

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Feb 08 2001

A060057 Nonzero numbers in expansion of ((tan(x))^5)/5! in (x^n)/n!.

Original entry on oeis.org

1, 70, 5376, 508640, 59835776, 8658773760, 1519012888576, 318434742599680, 78726332782411776, 22686646587991654400, 7541693844185862373376, 2865717052710927775825920, 1234612260068935283167461376, 598705789750919858580870922240
Offset: 0

Views

Author

Wolfdieter Lang, Mar 16 2001

Keywords

Crossrefs

A059419 (fifth column without zeros), A059420, A060056.

Programs

  • PARI
    a059419(n, k) = n!*polcoef(tan(x+x*O(x^n))^k/k!, n);
a(n) = a059419(2*n+5, 5); \\ Seiichi Manyama, May 11 2025

Formula

a(n) = A059419(5+2*n,5).
Previous Showing 11-15 of 15 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.