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 A155100 Triangle read by rows: coefficients in polynomials P_n(u) arising from the expansion of D^(n-1) (tan x) in increasing powers of tan x for n>=1 and 1 for n=0. 17
 1, 0, 1, 1, 0, 1, 0, 2, 0, 2, 2, 0, 8, 0, 6, 0, 16, 0, 40, 0, 24, 16, 0, 136, 0, 240, 0, 120, 0, 272, 0, 1232, 0, 1680, 0, 720, 272, 0, 3968, 0, 12096, 0, 13440, 0, 5040, 0, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320, 7936, 0, 176896, 0, 814080, 0, 1491840 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS The definition is d^(n-1) tan x / dx^n = P_n(tan x) for n>=1 and 1 for n=0. Interpolates between factorials and tangent numbers. From Peter Bala, Mar 02 2011: (Start) Companion triangles are A104035 and A185896. A combinatorial interpretation for the polynomial P_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 and put x_0 = -(n+1) and x_(n+1) = (-1)^n*(n+1). Then x_0,x_1,...,x_n,x_(n+1) is a snake of type S(n) when x_0 < x_1 > x_2 < ... x_(n+1). For example, -5 4 -3 -1 -2 5 is a snake of type S(4). Let sc be the number of sign changes through a snake sc = #{i, 0 <= i <= n, x_i*x_(i+1) < 0}. For example, the snake -5 4 -3 -1 -2 5 has sc = 3. The polynomial P_(n+1)(t) is the generating function for the sign change statistic on snakes of type S(n): P_(n+1)(t) = sum {snakes in S(n)} t^sc. See the example section below for the cases n=1 and n=2. (End) REFERENCES R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287. LINKS G. C. Greubel, Rows n=0..101 of triangle, 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. Gordon Haigh, A "natural" approach to Pick's theorem, Math. Gaz. 64 (1980), no. 429, 173-180. Michael E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23-30. Michael E. Hoffman, Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p. Donald E. Knuth and Thomas J. Buckholtz, Computation of tangent, Euler and Bernoulli numbers, Math. Comp. 21 1967 663-688. FORMULA If the polynomials are denoted by P_n(u), we have the recurrence P_{-1}=1, P_0 = u, P_n = (u^2+1)*dP_{n-1}/du. G.f.: Sum_{n >= 0} P_n(u) t^n/n! = (sin t + u*cos t)/(cos t - u sin t). [Hoffman] From Peter Bala, Feb 07 2011: (Start) RELATION WITH BERNOULLI NUMBERS A000367 AND A002445 Put T(n,t) = P_n(i*t), where i = sqrt(-1). We have the definite integral evaluation, valid when both m and n are >=1 and m+n >= 4: int( T(m,t)*T(n,t)/(1-t^2), t = -1..1) = (-1)^((m-n)/2)*2^(m+n-1)*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. RELATION WITH OTHER ROW POLYNOMIALS The following three identities hold for n >= 1: P_(n+1)(t) = (1+t^2)*R(n-1,t) where R(n,t) is the n-th row polynomial of A185896. P_(n+1)(t) = (-2*i)^n*(t-i)*R(n,-1/2+1/2*i*t), where i = sqrt(-1) and R(n,x) is an ordered Bell polynomial, that is, the n-th row polynomial of A019538. P_(n+1)(t) = (t-i)*(t+i)^n*A(n,(t-i)/(t+i)), 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 - see A008292. (End) EXAMPLE The polynomials P_{-1}(u) through P_6(u) with exponents in decreasing order: 1 u u^2+1 2*u^3+2*u 6*u^4+8*u^2+2 24*u^5+40*u^3+16*u 120*u^6+240*u^4+136*u^2+16 720*u^7+1680*u^5+1232*u^3+272*u Triangle begins: 1 0,1 1,0,1 0,2,0,2 2,0,8,0,6 0,16,0,40,0,24 16,0,136,0,240,0,120 0,272,0,1232,0,1680,0,720 272,0,3968,0,12096,0,13440,0,5040 0,7936,0,56320,0,129024,0,120960,0,40320 7936,0,176896,0,814080,0,1491840,0,1209600,0,362880 0,353792,0,3610112,0,12207360,0,18627840,0,13305600,0,3628800 353792,0,11184128,0,71867136,0,191431680,0,250145280,0,159667200,0,39916800 0,22368256,0,309836800,0,1436058624,0,3149752320,0,3597834240,0,2075673600, 0,479001600 From Peter Bala, Feb 07 2011: (Start) Examples of sign change statistic sc on snakes of type S(n): .....Snakes....# sign changes sc.......t^sc = = = = = = = = = = = = = = = = = = = = = = n=1 ...-2 1 -2............2................t^2 ...-2 -1 -2...........0................1 yields P_2(t) = 1+t^2; n=2 ...-3 1 -2 3 .........3................t^3 ...-3 2 1 3 ..........1................t ...-3 2 -1 3 .........3................t^3 ...-3 -1 -2 3 ........1................t yields P_3(t) = 2*t+2*t^3 (End). MAPLE P:=proc(n) option remember; if n=-1 then RETURN(1); elif n=0 then RETURN(u); else RETURN(expand((u^2+1)*diff(P(n-1), u))); fi; end; for n from -1 to 12 do t1:=series(P(n), u, 20); lprint(seriestolist(t1)); od: # Alternatively: with(PolynomialTools): seq(print(CoefficientList(`if`(i=0, 1, D@@(i-1))(tan), tan)), i=0..7); # Peter Luschny, May 19 2015 MATHEMATICA p[n_, u_] := D[Tan[x], {x, n}] /. Tan[x] -> u /. Sec[x] -> Sqrt[1 + u^2] // Expand; p[-1, u_] = 1; Flatten[ Table[ CoefficientList[ p[n, u], u], {n, -1, 9}]] (* Jean-François Alcover, Jun 28 2012 *) CROSSREFS For other versions of this triangle see A008293, A101343. A104035 is a companion triangle. Highest order coefficients give factorials A000142. Constant terms give tangent numbers A000182. Other coefficients: A002301. Setting u=1 in P_n gives A000831, u=2 gives A156073, u=3 gives A156075, u=4 gives A156076, u=1/2 gives A156102. Setting u=sqrt(2) in P_n gives A156108 and A156122; setting u=sqrt(3) gives A156103 and A000436. Cf. A008292, A019538, A185896. Sequence in context: A252706 A139137 A138231 * A076880 A082115 A161553 Adjacent sequences:  A155097 A155098 A155099 * A155101 A155102 A155103 KEYWORD nonn,tabl,nice AUTHOR N. J. A. Sloane, Nov 05 2009 EXTENSIONS Name clarified by Peter Luschny, May 25 2015 STATUS approved

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Last modified October 20 15:15 EDT 2019. Contains 328267 sequences. (Running on oeis4.)