

A198895


Triangle of coefficients arising in expansion of nth derivative of tan(x) + sec(x).


0



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
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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 ShiMei Ma (2011, 2012).
(End)


REFERENCES

Florence Nightingale David and D. E. Barton, Combinatorial Chance, Charles Griffin, 1962; see pp. 159162.


LINKS

Table of n, a(n) for n=0..51.
ShiMei Ma, Derivative polynomials and permutations by numbers of interior peaks and left peaks, arXiv:1106.5781 [math.CO], 2011.
ShiMei Ma, Derivative polynomials and enumeration of permutations by number of interior and left peaks , Discrete Mathematics 312(2) (2012), 405412.
Wikipedia, Florence Nightingale David.


FORMULA

nth 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(n1, k) + (n  k + 1) * T(n1, k2) for n >= 0 and 2 <= k <= n with initial conditions T(n, k=0) = 1 for n >= 0, T(n, k=1) = 2^(n1) 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, k1) + 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 * A118686 A102610 A203300
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



