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
Examples
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
...
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.
Crossrefs
Cf. A185410.
Programs
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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 */
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
Extensions
More terms from Franck Maminirina Ramaharo, Feb 05 2019