A198895 Triangle of coefficients arising in expansion of n-th derivative of tan(x) + sec(x).

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, 272, 61, 1, 64, 543, 1824, 3111, 2880, 1385, 272, 1, 128, 1636, 7680, 18270, 24576, 19028, 7936, 1385, 1, 256, 4916, 31616, 101166, 185856, 206276

Offset: 0

Author

N. J. A. Sloane, Oct 31 2011

Keywords

• nonn
• tabl

Comments

From Petros Hadjicostas, Aug 10 2019: (Start)

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).

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). (End)

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  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    272     61
  1  64  543  1824   3111   2880   1385    272
  1 128 1636  7680  18270  24576  19028   7936  1385
  1 256 4916 31616 101166 185856 206276 137216 50521 7936
  ...
		

References

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

Links

• Shi-Mei Ma, Derivative polynomials and permutations by numbers of interior peaks and left peaks, arXiv:1106.5781 [math.CO], 2011.
• Shi-Mei Ma, Derivative polynomials and enumeration of permutations by number of interior and left peaks , Discrete Mathematics 312(2) (2012), 405-412.
• Wikipedia, Florence Nightingale David.

Crossrefs

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

Formula

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).

From Petros Hadjicostas, Aug 10 2019: (Start)

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.

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.

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). (End)

Extensions

More terms from Max Alekseyev, Feb 17 2012