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A256978
Irregular triangle read by rows: coefficients of polynomials related to Stirling permutations.
0
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
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
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Apr 23 2015
EXTENSIONS
More terms from Franck Maminirina Ramaharo, Feb 05 2019
STATUS
approved