Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #34 Aug 05 2024 14:03:08
%S 1,1,1,1,4,1,11,4,1,26,34,1,57,180,34,1,120,768,496,1,247,2904,4288,
%T 496,1,502,10194,28768,11056,1,1013,34096,166042,141584,11056,1,2036,
%U 110392,868744,1372088,349504,1,4083,349500,4247720,11204160,6213288,349504
%N Triangle read by rows: number of simsun n-permutations with k descents.
%C Is this A094503 after removal of the top row? - _R. J. Mathar_, Aug 13 2008
%C Yes. See formula of _Peter Bala_, Jun 26 2012 in A094503. - _Stefano Spezia_, Aug 09 2023
%H Chak-On Chow and Wai Chee Shiu, <a href="https://doi.org/10.1007/s00026-011-0113-6">Counting Simsun Permutations by Descents</a>, Ann. Comb. 15, 625-635 (2011). See p. 627.
%H Ming-Jian Ding and Bao-Xuan Zhu, <a href="https://doi.org/10.1016/j.aam.2023.102591">Some results related to Hurwitz stability of combinatorial polynomials</a>, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 35.
%H R. P. Stanley, <a href="https://math.mit.edu/~rstan/pubs/pubfiles/96.pdf">Flag f-vectors and the cd-index</a>, Math. Zeitschrift 216 (1994), 483-499.
%H S. Sundaram, <a href="http://citeseerx.ist.psu.edu/pdf/08428845a2d7735bf0d5467a3b51ef55de495e3b">Plethysm, Partitions with an Even Number of Blocks and Euler Numbers</a>, in "Formal Power Series and Algebraic Combinatorics 1994," DIMACS Series in Discrete Mathematics and Theoretical Computer Science 24, AMS (1996).
%F T(n, k) = (k+1)*T(n-1, k) + (n-2k+1)*T(n-1, k-1);
%F Row g.f.: T(n, t) = Sum_{k=0..floor(n/2)} T(n, k)*t^k,
%F T(n, t) = ((n-1)*t + 1)*T(n-1, t) + t*(1-2t)*T(n-1, t)'.
%F E.g.f.: Sum_{n>=1} T(n, t)*x^n/n! = (2t-1)*(sec(x*sqrt(2t-1)/2)/(sqrt(2t-1) - tan(x*sqrt(2t-1)/2)))^2.
%e Triangle begins
%e 1;
%e 1, 1;
%e 1, 4;
%e 1, 11, 4;
%e 1, 26, 34;
%e 1, 57, 180, 34;
%e ...
%t Table[SeriesCoefficient[(2t-1)*(Sec[x*Sqrt[2t-1]/2]/(Sqrt[2t-1]- Tan[x*Sqrt[2t-1]/2]))^2,{x,0,n},{t,0,k}]n!,{n,11},{k,0,Floor[n/2]}]//Flatten (* _Stefano Spezia_, Aug 09 2023 *)
%Y Cf. A000111, A000295, A002105.
%K easy,nonn,tabf
%O 1,5
%A Chak-On Chow (cchow(AT)alum.mit.edu), Jan 28 2006
%E Corrected and extended by _Vladeta Jovovic_, Jan 30 2006