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 A185896 Triangle of coefficients of (1/sec^2(x))*D^n(sec^2(x)) in powers of t = tan(x), where D = d/dx. 10
 1, 0, 2, 2, 0, 6, 0, 16, 0, 24, 16, 0, 120, 0, 120, 0, 272, 0, 960, 0, 720, 272, 0, 3696, 0, 8400, 0, 5040, 0, 7936, 0, 48384, 0, 80640, 0, 40320, 7936, 0, 168960, 0, 645120, 0, 846720, 0, 362880, 0, 353792, 0, 3256320, 0, 8951040, 0, 9676800, 0, 3628800 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS DEFINITION Define polynomials R(n,t) with t = tan(x) by ... (d/dx)^n sec^2(x) = R(n,tan(x))*sec^2(x). The first few are ... R(0,t) = 1 ... R(1,t) = 2*t ... R(2,t) = 2 + 6*t^2 ... R(3,t) = 16*t + 24*t^3. This triangle shows the coefficients of R(n,t) in ascending powers of t called the tangent number triangle in [Hodges and Sukumar]. The polynomials R(n,t) form a companion polynomial sequence to Hoffman's two polynomial sequences - P(n,t) (A155100), the derivative polynomials of the tangent and Q(n,t) (A104035), the derivative polynomials of the secant. See also A008293 and A008294. COMBINATORIAL INTERPRETATION A combinatorial interpretation for the polynomial R(n,t) as the generating function for a sign change statistic on certain types of signed permutation can be found in [Verges]. A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...|x_n|} = {1,2...,n}. They form a group, the hyperoctahedral group of order 2^n*n! = A000165(n), isomorphic to the group of symmetries of the n dimensional cube. Let x_1,...,x_n be a signed permutation. Then 0,x_1,...,x_n,0 is a snake of type S(n;0,0) when 0 < x_1 > x_2 < ... 0. For example, 0 4 -3 -1 -2 0 is a snake of type S(4;0,0). Let sc be the number of sign changes through a snake ... sc = #{i, 1 <= i <= n-1, x_i*x_(i+1) < 0}. For example, the snake 0 4 -3 -1 -2 0  has sc = 1. The polynomial R(n,t) is the generating function for the sign change statistic on snakes of type S(n+1;0,0): ... R(n,t) = sum {snakes in S(n+1;0,0)} t^sc. See the example section below for the cases n=1 and n=2. PRODUCTION MATRIX Define three arrays R, L, and S as ... R = superdiag[2,3,4,...] ... L = subdiag[1,2,3,...] ... S = diag[2,4,6,...] with the indicated sequences on the main superdiagonal, the main subdiagonal and main diagonal, respectively, and 0's elsewhere. The array R+L is the production array for this triangle: the first row of (R+L)^n produces the n-th row of the triangle. On the vector space of complex polynomials the array R, the raising operator, represents the operator p(x) - > d/dx (x^2*p(x)), and the array L, the lowering operator, represents the differential operator d/dx - see Formula (4) below. The three arrays satisfy the commutation relations ... [R,L] = S, [R,S] = 2*R, [L,S] = -2*L and hence give a representation of the Lie algebra sl(2). LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened K. Boyadzhiev, Derivative Polynomials for tanh, tan, sech and sec in Explicit Form, arXiv:0903.0117 [math.CA], 2009-2010. M-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005. A. Hodges and C. V. Sukumar, Bernoulli, Euler, permutations and quantum algebras, Proc. R. Soc. A (2007) 463, 2401-2414 doi:10.1098/rspa.2007.0001 Michael E. Hoffman, Derivative polynomials, Euler polynomials,and associated integer sequences, Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R21. Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30. M. J-Verges, Enumeration of snakes and cycle-alternating permutations, arXiv:1011.0929 [math.CO], 2010. FORMULA GENERATING FUNCTION E.g.f.: (1)... F(t,z) = 1/(cos(z)-t*sin(z))^2 = sum {n>=0} R(n,t)*z^n/n! = 1 +  (2*t)*z + (2+6*t^2)*z^2/2! + (16*t+24*t^3)*z^3/3! + .... The e.g.f. equals the square of the e.g.f. of A104035. Continued fraction representation for the o.g.f: (2)... F(t,z) = 1/(1-2*t*z - 2*(1+t^2)*z^2/(1-4*t*z -...- n*(n+1)*(1+t^2) *z^2/(1-2*n*(n+1)*t*z -.... RECURRENCE RELATION (3)... T(n,k) = (k+1)*(T(n-1,k-1) + T(n-1,k+1)). ROW POLYNOMIALS The polynomials R(n,t) satisfy the recurrence relation (4)... R(n+1,t) = d/dt{(1+t^2)*R(n,t)} with R(0,t) = 1. Let D be the derivative operator d/dt and U = t, the shift operator. (5)... R(n,t) = (D + DUU)^n 1 RELATION WITH OTHER SEQUENCES A) Derivative Polynomials A155100 The polynomials (1+t^2)*R(n,t) are the  polynomials P_(n+2)(t) of B) Bernoulli Numbers A000367 and A002445 Put S(n,t) = R(n,i*t), where i = sqrt(-1). We have the definite integral evaluation (6)... int((1-t^2)*S(m,t)*S(n,t), t = -1..1) = (-1)^((m-n)/2)*2^(m+n+3) *Bernoulli(m+n+2). The case  m = n is equivalent to the result of [Grosset and Veselov]. The methods used there extend to the general case. C) Zigzag Numbers A000111 (7)... R_n(1) = A000828(n+1) = 2^n*A000111(n+1). D) Eulerian Numbers A008292 The polynomials R(n,t) are related to the Eulerian polynomials A(n,t) via (8)... R(n,t) = (t+i)^n*A(n+1,(t-i)/(t+i)) with the inverse identity (9)... A(n+1,t) = (-i/2)^n*(1-t)^n*R(n,i*(1+t)/(1-t)), where {A(n,t)}n>=1 = [1,1+t,1+4*t+t^2,1+11*t+11*t^2+t^3,...] is the sequence of Eulerian polynomials and i = sqrt(-1). E) Ordered set partitions A019538 (10)... R(n,t) = (-2*i)^n*T(n+1,x)/x, where x = i/2*t - 1/2 and T(n,x) is the n-th row po1ynomial of A019538; F) Miscellaneous Column 1 is the sequence of tangent numbers - see A000182. A000670(n+1) = (-i/2)^n*R(n,3*i). A004123(n+2) = 2*(-i/2)^n*R(n,5*i). A080795(n+1) =(-1)^n*(sqrt(-2))^n*R(n,sqrt(-2)). - Peter Bala, Aug 26 2011 From Leonid Bedratyuk, Aug 12 2012: (Start) T(n,k) = (-1)^(n+1)*(-1)^((n-k)/2)*Sum_{j=k+1..n+1} j! *stirling2(n+1,j) *2^(n+1-j) *(-1)^(n+j-k) *binomial(j-1,k)), see A059419. Sum_{j=i+1..n+1}((1-(-1)^(j-i))/(2*(j-i))*(-1)^((n-j)/2)*T(n,j))=(n+1)*(-1)^((n-1-i)/2)*T(n-1,i), for n>1 and  0n, 0, if(n==k, n!, (k+1)*(T(n-1, k-1) + T(n-1, k+1))))}; (PARI) {T(n, k) = my(A); if( n<0 || k>n, 0, A=1; for(i=1, n, A = ((1 + x^2) * A)'); polcoeff(A, k))}; /* Michael Somos, Jun 24 2017 */ CROSSREFS Cf. A000182, A000364, A000828 (row sums), A008292, A008293, A008294, A104035, A155100. Sequence in context: A221396 A221337 A157077 * A076256 A127467 A271708 Adjacent sequences:  A185893 A185894 A185895 * A185897 A185898 A185899 KEYWORD nonn,easy,tabl AUTHOR Peter Bala, Feb 07 2011 STATUS approved

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Last modified December 14 15:01 EST 2019. Contains 329979 sequences. (Running on oeis4.)