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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!.

%I A059419 #78 Feb 28 2020 08:18:07
%S A059419 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,
%T A059419 3968,0,2016,0,112,0,1,7936,0,28160,0,5376,0,168,0,1,0,176896,0,
%U A059419 135680,0,12432,0,240,0,1,353792,0,1805056,0,508640,0,25872,0,330,0,1,0
%N 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!.
%C A059419 (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
%C A059419 Also the Bell transform of A009006(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016
%D A059419 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259.
%H A059419 Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x))</a>.
%H A059419 Vladimir Kruchinin, <a href="https://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H A059419 Toufik Mansour, Mark Shattuck, <a href="https://doi.org/10.22108/toc.2017.102359.1483">Combinatorial parameters on bargraphs of permutations</a>, Transactions on Combinatorics, Article 1, Vol. 7, Issue 2, June 2018, Page 1-16.
%F A059419 T(n+1, k) = T(n, k-1) + k*(k+1)*T(n, k+1), T(n, n) = 1.
%F A059419 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
%F A059419 E.g.f.: exp(t*tan(x))-1 = t*x + t^2*x^2/2! + (2*t + t^3)*x^3/3! + ....
%F A059419 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
%F A059419 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
%F A059419 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
%e A059419      1;
%e A059419      0,     1;
%e A059419      2,     0,     1;
%e A059419      0,     8,     0,    1;
%e A059419     16,     0,    20,    0,    1;
%e A059419      0,   136,     0,   40,    0,   1;
%e A059419    272,     0,   616,    0,   70,   0,   1;
%e A059419      0,  3968,     0, 2016,    0, 112,   0,  1;
%e A059419   7936,     0, 28160,    0, 5376,   0, 168,  0,  1;
%p A059419 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
%p A059419 # The function BellMatrix is defined in A264428.
%p A059419 # Adds (1, 0, 0, 0, ..) as column 0.
%p A059419 BellMatrix(n -> 2^(n+1)*abs(euler(n+1, 1)), 10); # _Peter Luschny_, Jan 26 2016
%t A059419 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_ *)
%t A059419 rows = 12;
%t A059419 t = Table[2^(n+1)*Abs[EulerE[n+1, 1]], {n, 0, rows}];
%t A059419 T[n_, k_] := BellY[n, k, t];
%t A059419 Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o A059419 (PARI) T(n,k)=if(k<1 || k>n,0,n!*polcoeff(tan(x+x*O(x^n))^k/k!,n))
%o A059419 (Sage)
%o A059419 def A059419_triangle(dim):
%o A059419     M = matrix(ZZ, dim, dim)
%o A059419     for n in (0..dim-1): M[n,n] = 1
%o A059419     for n in (1..dim-1):
%o A059419         for k in (0..n-1):
%o A059419             M[n,k] = M[n-1,k-1]+(k+1)*(k+2)*M[n-1,k+1]
%o A059419     return M
%o A059419 A059419_triangle(9) # _Peter Luschny_, Sep 19 2012
%Y A059419 Diagonals give A000182, A024283, A059420 (interspersed with 0's), also A007290, A059421. Row sums give A006229. Essentially the same triangle as A008308.
%Y A059419 A111593 (signed triangle with extra column k=0 and row n=0).
%K A059419 nonn,easy,nice,tabl
%O A059419 1,4
%A A059419 _N. J. A. Sloane_, Jan 30 2001
%E A059419 More terms from Larry Reeves (larryr(AT)acm.org), Feb 01 2001