A256978 Irregular triangle read by rows: coefficients of polynomials related to Stirling permutations.

1, 1, 1, 1, 1, 3, 7, 3, 1, 1, 7, 29, 31, 29, 7, 1, 1, 15, 101, 195, 321, 195, 101, 15, 1, 1, 31, 327, 1001, 2507, 2661, 2507, 1001, 327, 31, 1, 1, 63, 1023, 4641, 16479, 26481, 37759, 26481, 16479, 4641, 1023, 63, 1, 1, 127, 3145, 20343, 98289, 221775, 439105, 461455, 439105, 221775, 98289, 20343, 3145, 127, 1

Offset

1,6

Links

Table of n, a(n) for n=1..64.

Shi-Mei Ma and Toufik Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, arXiv preprint, arXiv:1409.6525 [math.CO], 2014. See polynomials C_n(x).

Shi-Mei Ma and Toufik Mansour, The 1/k-Eulerian polynomials and k-Stirling permutations, Discrete Mathematics, Volume 338, Issue 8, 2015, 1468-1472.

Shi-Mei Ma, Yeong-Nan Yeh, The alternating run polynomials of permutations, arXiv:1904.11437 [math.CO], 2019. See p. 9.

Formula

E.g.f.: (exp(z*(x - 1)*(x + 1)) + x)/(x + 1)*sqrt((1 - x^2)/(exp(2*z*(x - 1)*(x + 1)) - x^2)) - 1. - Franck Maminirina Ramaharo, Feb 05 2019

Example

Triangle begins:
  n\k | 1   2   3    4    5    6    7    8   9  10  11
  ----+-----------------------------------------------
    1 | 1
    2 | 1   1   1
    3 | 1   3   7    3    1
    4 | 1   7  29   31   29    7    1
    5 | 1  15 101  195  321  195  101   15   1
    6 | 1  31 327 1001 2507 2661 2507 1001 327  31   1
    ...

Prog

(Maxima)
gf : taylor((exp(z*(x - 1)*(x + 1)) + x)/(x + 1)*sqrt((1 - x^2)/(exp(2*z*(x - 1)*(x + 1)) - x^2)) - 1, z, 0, 50)$
row(x, n) := n!*ratcoef(gf, z, n)$
create_list(ratcoef(row(x, n), x, k), n, 1, 20, k, 1, hipow(row(x, n), x));
/* Franck Maminirina Ramaharo, Feb 05 2019 */

Crossrefs

Cf. A185410.

Sequence in context: A133368 A153027 A161127 * A296442 A021272 A135613

Adjacent sequences: A256975 A256976 A256977 * A256979 A256980 A256981

Keyword

nonn,tabf

Author

N. J. A. Sloane, Apr 23 2015

Extensions

More terms from Franck Maminirina Ramaharo, Feb 05 2019

Status

approved