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

0,5

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)

References

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

Links

Table of n, a(n) for n=0..51.

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.

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)

Example

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
  ...

Crossrefs

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

Sequence in context: A308905 A158471 A158472 * A355635 A118686 A355540

Adjacent sequences: A198892 A198893 A198894 * A198896 A198897 A198898

Keyword

nonn,tabl

Author

N. J. A. Sloane, Oct 31 2011

Extensions

More terms from Max Alekseyev, Feb 17 2012

Status

approved