A059419 Triangle T(n,k) (1 <= k <= n) of tangent numbers, read by rows: T(n,k) = coefficient of x^n/n! in expansion of (tan x)^k/k!.

1, 0, 1, 2, 0, 1, 0, 8, 0, 1, 16, 0, 20, 0, 1, 0, 136, 0, 40, 0, 1, 272, 0, 616, 0, 70, 0, 1, 0, 3968, 0, 2016, 0, 112, 0, 1, 7936, 0, 28160, 0, 5376, 0, 168, 0, 1, 0, 176896, 0, 135680, 0, 12432, 0, 240, 0, 1, 353792, 0, 1805056, 0, 508640, 0, 25872, 0, 330, 0, 1, 0

Offset

1,4

Comments

(tan(x))^k = sum{n>0, If n+k is odd, T(n,k) = 0 = n!/k!*(-1)^((n+k)/2)*sum{j=k..n} (j!/n!) * Stirling2(n,j) * 2^(n-j) * (-1)^(n+j-k) * binomial(j-1,k-1)*x^n}. - Vladimir Kruchinin, Aug 13 2012

Also the Bell transform of A009006(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

References

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

Links

Table of n, a(n) for n=1..67.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).

Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Toufik Mansour, Mark Shattuck, Combinatorial parameters on bargraphs of permutations, Transactions on Combinatorics, Article 1, Vol. 7, Issue 2, June 2018, Page 1-16.

Formula

T(n+1, k) = T(n, k-1) + k*(k+1)*T(n, k+1), T(n, n) = 1.

If n+k is odd, T(n,k) = 0 = 1/k!*(-1)^((n+k)/2)*Sum_{j=k..n} j!* Stirling2(n,j)*2^(n-j)*(-1)^(n+j-k)*binomial(j-1,k-1). - Vladimir Kruchinin, Feb 10 2011

E.g.f.: exp(t*tan(x))-1 = t*x + t^2*x^2/2! + (2*t + t^3)*x^3/3! + ....

The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x^2)*d/dx. - Peter Bala, Nov 25 2011

The o.g.f.s of the diagonals of this triangle are rational functions obtained from 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! + .... For example, the fourth subdiagonal has o.g.f. 8*t*(2+3*t)/(1-t)^7 = 16*t + 136*t^2 + 616*t^3 + .... - Peter Bala, Apr 23 2012

With offset 0 and initial column of zeros, except for T(0,0) = 1, e.g.f.(t,x) = e^(x*tan(t)) = e^(P(.,x)t) ; the lowering operator, L = atan(d/dx) ; and the raising operator, R = x [1 +(d/dx)^2], where L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x). The sequence is a binomial Sheffer sequence. - Tom Copeland, Oct 01 2015

Example

     1;
     0,     1;
     2,     0,     1;
     0,     8,     0,    1;
    16,     0,    20,    0,    1;
     0,   136,     0,   40,    0,   1;
   272,     0,   616,    0,   70,   0,   1;
     0,  3968,     0, 2016,    0, 112,   0,  1;
  7936,     0, 28160,    0, 5376,   0, 168,  0,  1;

Maple

A059419 := proc(n, k) option remember; if n = k then 1; elif k <0 or k > n then 0; else  procname(n-1, k-1)+k*(k+1)*procname(n-1, k+1) ; end if; end proc: # R. J. Mathar, Feb 11 2011
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> 2^(n+1)*abs(euler(n+1, 1)), 10); # Peter Luschny, Jan 26 2016

Mathematica

d[f_ ] := (1+x^2)*D[f, x]; d[ f_, n_] := Nest[d, f, n]; row[n_] := Rest[ CoefficientList[ d[Exp[x*t], n] /. x -> 0, t]]; Flatten[ Table[ row[n], {n, 1, 12}]] (* Jean-François Alcover, Dec 21 2011, after Peter Bala *)
rows = 12;
t = Table[2^(n+1)*Abs[EulerE[n+1, 1]], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

Prog

(PARI) T(n, k)=if(k<1 || k>n, 0, n!*polcoeff(tan(x+x*O(x^n))^k/k!, n))
(Sage)
def A059419_triangle(dim):
    M = matrix(ZZ, dim, dim)
    for n in (0..dim-1): M[n, n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n, k] = M[n-1, k-1]+(k+1)*(k+2)*M[n-1, k+1]
    return M
A059419_triangle(9) # Peter Luschny, Sep 19 2012

Crossrefs

Diagonals give A000182, A024283, A059420 (interspersed with 0's), also A007290, A059421. Row sums give A006229. Essentially the same triangle as A008308.

A111593 (signed triangle with extra column k=0 and row n=0).

Sequence in context: A095403 A011328 A048277 * A185415 A049218 A212358

Adjacent sequences: A059416 A059417 A059418 * A059420 A059421 A059422

Keyword

nonn,easy,nice,tabl

Author

N. J. A. Sloane, Jan 30 2001

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Feb 01 2001

Status

approved