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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.

%I A155100 #81 Aug 16 2024 20:53:56
%S A155100 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,
%T A155100 272,0,1232,0,1680,0,720,272,0,3968,0,12096,0,13440,0,5040,0,7936,0,
%U A155100 56320,0,129024,0,120960,0,40320,7936,0,176896,0,814080,0,1491840
%N 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.
%C A155100 The definition is d^(n-1) tan x / dx^n = P_n(tan x) for n>=1 and 1 for n=0.
%C A155100 Interpolates between factorials and tangent numbers.
%C A155100 From _Peter Bala_, Mar 02 2011: (Start)
%C A155100 Companion triangles are A104035 and A185896.
%C A155100 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].
%C A155100 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}.
%C A155100 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.
%C A155100 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).
%C A155100 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.
%C A155100 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.
%C A155100 See the example section below for the cases n=1 and n=2.
%C A155100 (End)
%C A155100 Equals A107729 when the first column is removed. - _Georg Fischer_, Jul 26 2023
%D A155100 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998, p. 287.
%H A155100 G. C. Greubel, <a href="/A155100/b155100.txt">Rows n = 0..101 of triangle, flattened</a>
%H A155100 K. Boyadzhiev, <a href="http://arxiv.org/abs/0903.0117">Derivative Polynomials for tanh, tan, sech and sec in Explicit Form</a>, arXiv:0903.0117 [math.CA], 2009-2010.
%H A155100 M.-P. Grosset and A. P. Veselov, <a href="https://arxiv.org/abs/math/0503175">Bernoulli numbers and solitons</a>, arXiv:math/0503175 [math.GM], 2005.
%H A155100 Gordon Haigh, <a href="http://www.jstor.org/stable/3615119">A "natural" approach to Pick's theorem</a>, Math. Gaz. 64 (1980), no. 429, 173-180.
%H A155100 Michael E. Hoffman, <a href="http://www.jstor.org/stable/2974853">Derivative polynomials for tangent and secant</a>, Amer. Math. Monthly, 102 (1995), 23-30.
%H A155100 Michael E. Hoffman, <a href="https://doi.org/10.37236/1453">Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences</a>, The Electronic Journal of Combinatorics, Volume 6.1 (1999): Research paper R21, 13 p.
%H A155100 Donald E. Knuth and Thomas J. Buckholtz, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0221735-9">Computation of tangent, Euler and Bernoulli numbers</a>, Math. Comp. 21 1967 663-688.
%H A155100 Shi-Mei Ma, Qi Fang, Toufik Mansour, and Yeong-Nan Yeh, <a href="https://arxiv.org/abs/2104.09374">Alternating Eulerian polynomials and left peak polynomials</a>, arXiv:2104.09374 [math.CO], 2021.
%F A155100 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.
%F A155100 G.f.: Sum_{n >= 0} P_n(u) t^n/n! = (sin t + u*cos t)/(cos t - u sin t). [Hoffman]
%F A155100 From _Peter Bala_, Feb 07 2011: (Start)
%F A155100 RELATION WITH BERNOULLI NUMBERS A000367 AND A002445
%F A155100 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:
%F A155100 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).
%F A155100 The case m = n is equivalent to the result of [Grosset and Veselov]. The methods used there extend to the general case.
%F A155100 RELATION WITH OTHER ROW POLYNOMIALS
%F A155100 The following three identities hold for n >= 1:
%F A155100 P_(n+1)(t) = (1+t^2)*R(n-1,t) where R(n,t) is the n-th row polynomial of A185896.
%F A155100 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.
%F A155100 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)
%F A155100 T(n,k) = cos((n+k)*Pi/2) * Sum_{p=0..n-1} A008292(n-1,p+1) Sum_{j=0..k}(-1)^(p+j+1) * binomial(p+1,k-j) *binomial(n-p-1,j) for n>1. - _Ammar Khatab_, Aug 15 2024
%e A155100 The polynomials P_{-1}(u) through P_6(u) with exponents in decreasing order:
%e A155100       1
%e A155100       u
%e A155100       u^2 +    1
%e A155100     2*u^3 +    2*u
%e A155100     6*u^4 +    8*u^2 +    2
%e A155100    24*u^5 +   40*u^3 +   16*u
%e A155100   120*u^6 +  240*u^4 +  136*u^2 +  16
%e A155100   720*u^7 + 1680*u^5 + 1232*u^3 + 272*u
%e A155100   ...
%e A155100 Triangle begins:
%e A155100   1
%e A155100   0, 1
%e A155100   1, 0, 1
%e A155100   0, 2, 0, 2
%e A155100   2, 0, 8, 0, 6
%e A155100   0, 16, 0, 40, 0, 24
%e A155100   16, 0, 136, 0, 240, 0, 120
%e A155100   0, 272, 0, 1232, 0, 1680, 0, 720
%e A155100   272, 0, 3968, 0, 12096, 0, 13440, 0, 5040
%e A155100   0, 7936, 0, 56320, 0, 129024, 0, 120960, 0, 40320
%e A155100   7936, 0, 176896, 0, 814080, 0, 1491840, 0, 1209600, 0, 362880
%e A155100   0, 353792, 0, 3610112, 0, 12207360, 0, 18627840, 0, 13305600, 0, 3628800
%e A155100   ...
%e A155100 From _Peter Bala_, Feb 07 2011: (Start)
%e A155100 Examples of sign change statistic sc on snakes of type S(n):
%e A155100     Snakes     # sign changes sc  t^sc
%e A155100   ===========  =================  ====
%e A155100 n=1:
%e A155100   -2  1 -2 ........... 2 ........ t^2
%e A155100   -2 -1 -2 ........... 0 ........ 1
%e A155100                   yields P_2(t) = 1 + t^2;
%e A155100 n=2:
%e A155100   -3  1 -2  3 ........ 3 ........ t^3
%e A155100   -3  2  1  3 ........ 1 ........ t
%e A155100   -3  2 -1  3 ........ 3 ........ t^3
%e A155100   -3 -1 -2  3 ........ 1 ........ t
%e A155100                   yields P_3(t) = 2*t + 2*t^3. (End)
%p A155100 P:=proc(n) option remember;
%p A155100 if n=-1 then RETURN(1); elif n=0 then RETURN(u); else RETURN(expand((u^2+1)*diff(P(n-1),u))); fi;
%p A155100 end;
%p A155100 for n from -1 to 12 do t1:=series(P(n),u,20); lprint(seriestolist(t1)); od:
%p A155100 # Alternatively:
%p A155100 with(PolynomialTools): seq(print(CoefficientList(`if`(i=0,1,D@@(i-1))(tan),tan)), i=0..7); # _Peter Luschny_, May 19 2015
%t A155100 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 *)
%t A155100 T[ n_, k_] := Which[n<0, Boole[n==-1 && k==0], n==0, Boole[k==1], True, (k-1)*T[n-1, k-1] + (k+1)*T[n-1, k+1]]; (* _Michael Somos_, Jul 09 2024 *)
%o A155100 (PARI) {T(n, k) = if(n<0, n==-1 && k==0, n==0, k==1, (k-1)*T(n-1, k-1) + (k+1)*T(n-1, k+1))}; /* _Michael Somos_, Jul 09 2024 */
%Y A155100 For other versions of this triangle see A008293, A101343.
%Y A155100 A104035 is a companion triangle.
%Y A155100 Highest order coefficients give factorials A000142. Constant terms give tangent numbers A000182. Other coefficients: A002301.
%Y A155100 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.
%Y A155100 Setting u=sqrt(2) in P_n gives A156108 and A156122; setting u=sqrt(3) gives A156103 and A000436.
%Y A155100 Cf. A008292, A019538, A107729, A185896.
%K A155100 nonn,tabl,nice
%O A155100 0,8
%A A155100 _N. J. A. Sloane_, Nov 05 2009
%E A155100 Name clarified by _Peter Luschny_, May 25 2015