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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!. 16
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

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Last modified April 19 21:11 EDT 2021. Contains 343117 sequences. (Running on oeis4.)