This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A091156 #79 Apr 07 2025 16:07:19
%S A091156 1,1,1,1,1,4,1,11,2,1,26,15,1,57,69,5,1,120,252,56,1,247,804,364,14,1,
%T A091156 502,2349,1800,210,1,1013,6455,7515,1770,42,1,2036,16962,27940,11055,
%U A091156 792,1,4083,43086,95458,57035,8217,132,1,8178,106587,305812,257257
%N A091156 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having k long ascents (i.e., ascents of length at least 2). Rows are of length 1,1,2,2,3,3,... .
%C A091156 Also number of ordered trees with n edges, having k branch nodes (i.e., vertices of outdegree at least 2).
%C A091156 Also number of Łukasiewicz paths of length n having k fall steps (1,-1) that start at an odd level. A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: T(4,2)=2 because we have U(D)U(D) and U(3)(D)D(D), where U=(1,1), D=(1,-1), U(3)=(1,3) and the fall steps that start at an odd level are shown between parentheses. Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). T(2n,n)=A000108(n). T(2n+1,n)=A001791(n+1)=binomial(2n+2,n). - _Emeric Deutsch_, Jan 06 2005
%C A091156 Also number of Dyck paths of semilength n with k UUD's - I. Tasoulas (jtas(AT)unipi.gr), Feb 19 2006
%C A091156 T(n,k) = number of Dyck n-paths whose decomposition into 2-step subpaths contains k UUs. For example, T(4,2)=2 counts UU|UU|DD|DD, UU|DD|UU|DD (vertical bars indicate path decomposition). - _David Callan_, Jun 07 2006
%C A091156 T(n,k) = number of binary trees on n-1 edges containing k right edges whose child vertex has no right child. Under Knuth's "natural" correspondence, such a vertex in binary (n-1)-tree ~ a vertex of outdegree >=2 in ordered n-tree. - _David Callan_, Sep 25 2006
%C A091156 T(n,k) = number of binary trees on n-1 edges containing k left edges whose child vertex has no left child. Under "natural" correspondence, such a vertex in binary (n-1)-tree ~ a leaf edge with no left neighbor edge and not incident to the root in ordered n-tree ~ a UUD in Dyck n-path. - _David Callan_, Sep 25 2006
%C A091156 T(n,k) = number of permutations of length n avoiding 321 (classically) with k descents. - _Andrew Baxter_, May 17 2011.
%D A091156 R. P. Stanley, Enumerative Combinatorics, Vol. 1, 1986; See Exercise 3.71(f).
%H A091156 Alois P. Heinz, <a href="/A091156/b091156.txt">Rows n = 0..200, flattened</a>
%H A091156 M. Barnabei et al., <a href="http://arxiv.org/abs/0910.0963">The descent statistic over 123-avoiding permutations</a>, arXiv:0910.0963 [math.CO], 2009.
%H A091156 A. M. Baxter, <a href="https://pdfs.semanticscholar.org/2c5d/79e361d3aecb25c380402144177ad7cd9dc8.pdfindex.html">Algorithms for Permutation Statistics</a>, Ph. D. Dissertation, Rutgers University, May 2011. See p. 88.
%H A091156 Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, Teresa Wheeland, <a href="https://arxiv.org/abs/1812.07112">Distributions of Statistics over Pattern-Avoiding Permutations</a>, arXiv:1812.07112 [math.CO], 2018.
%H A091156 Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-Sorting Preimages of Permutation Classes</a>, arXiv:1809.03123 [math.CO], 2018.
%H A091156 Katie R. Gedeon, <a href="https://arxiv.org/abs/1610.05349">Kazhdan-Lusztig polynomials of thagomizer matroids</a>, arXiv:1610.05349 [math.CO], 2016.
%H A091156 Y. Park, S. Park, <a href="http://dx.doi.org/10.4134/JKMS.2013.50.3.529">Avoiding permutations and the Narayana numbers</a>, J. Korean Math. Soc. 50 (2013), No. 3, pp. 529-541.
%H A091156 Lara Pudwell, <a href="http://permutationpatterns.com/slides/Pudwell.pdf">On the distribution of peaks (and other statistics)</a>, 16th International Conference on Permutation Patterns, Dartmouth College, 2018.
%H A091156 A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924.
%H A091156 Chao-Jen Wang, <a href="http://people.brandeis.edu/~gessel/homepage/students/wangthesis.pdf">Applications of the Goulden-Jackson cluster method to counting Dyck paths by occurrences of subwords</a>.
%H A091156 Philip B. Zhang and Sergey Kitaev, <a href="https://doi.org/10.1016/j.dam.2025.03.032">Descent generating polynomials for (n-3)- and (n-4)-stack-sortable (pattern-avoiding) permutations</a>, Discrete Applied Mathematics, Volume 372, 2025, Pages 1-14.
%H A091156 <a href="/index/Lu#Lukasiewicz">Index entries for sequences related to Łukasiewicz</a>
%F A091156 T(n,k) = (1/(n+1)) * binomial(n+1, k) * Sum_{j=0..n-2k} binomial(k+j-1, k-1)*binomial(n+1-k, n-2k-j).
%F A091156 G.f. G(t, z) satisfies z*(1-z+t*z)*G^2 - G + 1 = 0.
%F A091156 T(n,k) = n!*(1+k)/((n-2*k)!*(1+k)!^2)*hypergeom([k,2*k-n], [k+2], -1). - _Peter Luschny_, Oct 16 2015
%F A091156 T(n,k) = A055151(n,k)*hypergeom([k,2*k-n],[k+2],-1). - _Peter Luschny_, Oct 16 2015
%e A091156 T(4,1) = 11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD (two long ascents) and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1).
%e A091156 Triangle begins:
%e A091156 1;
%e A091156 1;
%e A091156 1, 1;
%e A091156 1, 4;
%e A091156 1, 11, 2;
%e A091156 1, 26, 15;
%e A091156 1, 57, 69, 5;
%e A091156 1, 120, 252, 56;
%e A091156 1, 247, 804, 364, 14;
%e A091156 1, 502, 2349, 1800, 210;
%e A091156 1, 1013, 6455, 7515, 1770, 42;
%e A091156 ...
%p A091156 a := (n,k)->binomial(n+1,k)* add(binomial(k+j-1,k-1)*binomial(n+1-k, n-2*k-j), j=0..n-2*k)/(n+1); seq(seq(a(n,k), k=0..floor(n/2)),n=0..15);
%p A091156 seq(seq(simplify(n!*(1+k)/((n-2*k)!*(1+k)!^2)*hypergeom([k,2*k-n],[k+2],-1)),k=0.. floor(n/2)),n=0..15); # _Peter Luschny_, Oct 16 2015
%p A091156 # alternative Maple program:
%p A091156 b:= proc(x, y) option remember; `if`(y>x or y<0, 0,
%p A091156 `if`(x=0, 1, expand(b(x-1, y)*`if`(y=0, 1, 2)*z+
%p A091156 b(x-1, y+1) +b(x-1, y-1))))
%p A091156 end:
%p A091156 T:= n-> (p-> seq(coeff(p, z, n-2*i), i=0..n/2))(b(n, 0)):
%p A091156 seq(T(n), n=0..15); # _Alois P. Heinz_, Aug 07 2018
%t A091156 T[n_, k_] := Binomial[n+1, k]*Sum[Binomial[k+j-1, k-1]*Binomial[n+1-k, n- 2*k-j], {j, 0, n-2*k}]/(n+1); Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/2 ]}] // Flatten (* _Jean-François Alcover_, Jan 31 2016 *)
%o A091156 (PARI)
%o A091156 tabf(nn) = {for(n=-1, nn, for(k=0, floor(n/2), if(binomial(n+1,k) * sum(j=0, n-2*k, binomial(k+j-1,k-1) * binomial(n+1-k,n-2*k-j))/(n+1)==0,print1("1, "), print1(binomial(n+1,k) * sum(j=0, n-2*k, binomial(k+j-1,k-1) * binomial(n+1-k,n-2*k-j))/(n+1),", "));); print();); };
%o A091156 tabf(16); \\ _Indranil Ghosh_, Mar 05 2017
%Y A091156 Cf. A000108, A001791, A243752.
%Y A091156 T(n,k) are rational multiples of A055151.
%K A091156 nonn,tabf
%O A091156 0,6
%A A091156 _Emeric Deutsch_, Feb 22 2004
%E A091156 Edited by _Andrew Baxter_, May 17 2011