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Triangle of coefficients arising in expansion of n-th derivative of tan(x) + sec(x).
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%I #42 Aug 11 2019 02:35:37

%S 1,1,1,1,2,1,1,4,5,2,1,8,18,16,5,1,16,58,88,61,16,1,32,179,416,479,

%T 272,61,1,64,543,1824,3111,2880,1385,272,1,128,1636,7680,18270,24576,

%U 19028,7936,1385,1,256,4916,31616,101166,185856,206276

%N Triangle of coefficients arising in expansion of n-th derivative of tan(x) + sec(x).

%C From _Petros Hadjicostas_, Aug 10 2019: (Start)

%C The recurrence about T(n, k) and the equation that connects T(n, k) to P(n, k) = A059427(n,k), which are given below, appear on p. 159 of the book by David and Barton (1962). The initial conditions, however, for their triangular array S^*_{N,t} are slightly different, but there is an agreement starting at t = k = 1. They do not provide tables for S^*_{N,t) (that matches the current array T(n, k) for N = n >= 0 and t = k >= 1).

%C Despite the slightly different initial conditions between T(n, k) and S^*_{N,t} (from p. 159 in the book), the recurrence given below can be proved very easily from the recurrence for the row polynomials R_n(x) given in Shi-Mei Ma (2011, 2012).

%C (End)

%D Florence Nightingale David and D. E. Barton, Combinatorial Chance, Charles Griffin, 1962; see pp. 159-162.

%H Shi-Mei Ma, <a href="http://arxiv.org/abs/1106.5781">Derivative polynomials and permutations by numbers of interior peaks and left peaks</a>, arXiv:1106.5781 [math.CO], 2011.

%H Shi-Mei Ma, <a href="https://doi.org/10.1016/j.disc.2011.10.003">Derivative polynomials and enumeration of permutations by number of interior and left peaks </a>, Discrete Mathematics 312(2) (2012), 405-412.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Florence_Nightingale_David">Florence Nightingale David</a>.

%F n-th row represents the coefficients of the polynomial R_n(x) defined by the recurrence: R_0(x) = 1, R_1(x) = 1 + x, and for n >= 1, R_{n+1}(x) = (1 + n*x^2)*R_n(x) + x*(1 - x^2)*R'_n(x).

%F From _Petros Hadjicostas_, Aug 10 2019: (Start)

%F T(n, k) = (k + 1) * T(n-1, k) + (n - k + 1) * T(n-1, k-2) for n >= 0 and 2 <= k <= n with initial conditions T(n, k=0) = 1 for n >= 0, T(n, k=1) = 2^(n-1) for n >= 1, and T(n, k) = 0 for n < 0 or n < k.

%F Setting x = 1 in the equation R_{n+1}(x) = (1 + n*x^2)*R_n(x) + x*(1 - x^2)*R'_n(x) (valid for n >= 1), we get R_{n+1}(1) = (n + 1)*R_n(1) for n >= 1. Since R_1(1) = 2, we have that R_n(1) = 2*n! for n >= 1. Since also R_0(1) = 1, we conclude that Sum_{k = 0..n} T(n,k) = R_n(1) = 2*n! - 0^n = A098558(n) for n >= 0.

%F Let P(n, k) = A059427(n,k) with P(n, k) = 0 for n <= 1 or n <= k. Then T(n, k) = (1/2)*P(n, k-1) + P(n, k) + (1/2) * P(n, k+1) for n >= 2 and 0 <= k <= n (but this is not true for n = 0 and n = 1).

%F (End)

%e Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:

%e 1

%e 1 1

%e 1 2 1

%e 1 4 5 2

%e 1 8 18 16 5

%e 1 16 58 88 61 16

%e 1 32 179 416 479 272 61

%e 1 64 543 1824 3111 2880 1385 272

%e 1 128 1636 7680 18270 24576 19028 7936 1385

%e 1 256 4916 31616 101166 185856 206276 137216 50521 7936

%e ...

%Y Cf. A059427, A098558 (row sums), A000111 (diagonal and 1st subdiagonal), A000340 (column 3) A000431 (column 4), A000363 (column 5)

%K nonn,tabl

%O 0,5

%A _N. J. A. Sloane_, Oct 31 2011

%E More terms from _Max Alekseyev_, Feb 17 2012