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

%I A256978 #14 Apr 25 2019 23:13:51
%S A256978 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,
%T A256978 31,327,1001,2507,2661,2507,1001,327,31,1,1,63,1023,4641,16479,26481,
%U A256978 37759,26481,16479,4641,1023,63,1,1,127,3145,20343,98289,221775,439105,461455,439105,221775,98289,20343,3145,127,1
%N A256978 Irregular triangle read by rows: coefficients of polynomials related to Stirling permutations.
%H A256978 Shi-Mei Ma and Toufik Mansour, <a href="http://arxiv.org/abs/1409.6525">The 1/k-Eulerian polynomials and k-Stirling permutations</a>, arXiv preprint, arXiv:1409.6525 [math.CO], 2014. See polynomials C_n(x).
%H A256978 Shi-Mei Ma and Toufik Mansour, <a href="https://doi.org/10.1016/j.disc.2015.03.015">The 1/k-Eulerian polynomials and k-Stirling permutations</a>, Discrete Mathematics, Volume 338, Issue 8, 2015, 1468-1472.
%H A256978 Shi-Mei Ma, Yeong-Nan Yeh, <a href="https://arxiv.org/abs/1904.11437">The alternating run polynomials of permutations</a>, arXiv:1904.11437 [math.CO], 2019. See p. 9.
%F A256978 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
%e A256978 Triangle begins:
%e A256978   n\k | 1   2   3    4    5    6    7    8   9  10  11
%e A256978   ----+-----------------------------------------------
%e A256978     1 | 1
%e A256978     2 | 1   1   1
%e A256978     3 | 1   3   7    3    1
%e A256978     4 | 1   7  29   31   29    7    1
%e A256978     5 | 1  15 101  195  321  195  101   15   1
%e A256978     6 | 1  31 327 1001 2507 2661 2507 1001 327  31   1
%e A256978     ...
%o A256978 (Maxima)
%o A256978 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)$
%o A256978 row(x, n) := n!*ratcoef(gf, z, n)$
%o A256978 create_list(ratcoef(row(x, n), x, k), n, 1, 20, k, 1, hipow(row(x, n), x));
%o A256978 /* _Franck Maminirina Ramaharo_, Feb 05 2019 */
%Y A256978 Cf. A185410.
%K A256978 nonn,tabf
%O A256978 1,6
%A A256978 _N. J. A. Sloane_, Apr 23 2015
%E A256978 More terms from _Franck Maminirina Ramaharo_, Feb 05 2019