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%I #117 Apr 19 2024 11:30:40
%S 1,1,1,1,3,2,1,6,10,5,1,10,30,35,14,1,15,70,140,126,42,1,21,140,420,
%T 630,462,132,1,28,252,1050,2310,2772,1716,429,1,36,420,2310,6930,
%U 12012,12012,6435,1430,1,45,660,4620,18018,42042,60060,51480,24310,4862
%N Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.
%C Row sums: A006318 (Schroeder numbers). Essentially same as triangle A060693 transposed.
%C T(n,k) is number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k U's. E.g., T(2,1)=3 because we have UHD, UDH and HUD. - _Emeric Deutsch_, Dec 06 2003
%C Little Schroeder numbers A001003 have a(n) = Sum_{k=0..n} A088617(n,k)*(-1)^(n-k)*2^k. - _Paul Barry_, May 24 2005
%C Conjecture: The expected number of U's in a Schroeder n-path is asymptotically Sqrt[1/2]*n for large n. - _David Callan_, Jul 25 2008
%C T(n, k) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha) = |Dom(alpha)|). - _Abdullahi Umar_, Oct 02 2008
%C The antidiagonals of this lower triangular matrix are the rows of A055151. - _Tom Copeland_, Jun 17 2015
%D Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.
%H Michael De Vlieger, <a href="/A088617/b088617.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150)
%H Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, <a href="https://arxiv.org/abs/2301.09765">Enumeration of multi-rooted plane trees</a>, arXiv:2301.09765 [math.CO], 2023.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6.
%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H Paul Barry, <a href="https://arxiv.org/abs/2101.06713">On the inversion of Riordan arrays</a>, arXiv:2101.06713 [math.CO], 2021.
%H Manosij Ghosh Dastidar and Michael Wallner, <a href="https://arxiv.org/abs/2402.17849">Bijections and congruences involving lattice paths and integer compositions</a>, arXiv:2402.17849 [math.CO], 2024. See p. 16.
%H Samuele Giraudo, <a href="https://arxiv.org/abs/1903.00677">Tree series and pattern avoidance in syntax trees</a>, arXiv:1903.00677 [math.CO], 2019.
%H Hsien-Kuei Hwang and Satoshi Kuriki, <a href="https://arxiv.org/abs/2404.06040">Integrated empirical measures and generalizations of classical goodness-of-fit statistics</a>, arXiv:2404.06040 [math.ST], 2024. See p. 11.
%H C. Jordan, <a href="/A002457/a002457_1.pdf">Calculus of Finite Differences</a>, Budapest, 1939. [Annotated scans of pages 448-450 only]
%H A. Kirillov, <a href="https://arxiv.org/abs/1502.00426">On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials</a>, arXiv preprint arXiv:1502.00426 [math.RT], 2016.
%H M. Klazar, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00164-7">On numbers of Davenport-Schinzel sequences</a>, Discr. Math., 185 (1998), 77-87.
%H Paul W. Lapey and Aaron Williams, <a href="https://www.researchgate.net/profile/Aaron-Williams/publication/360053030_A_Shift_Gray_Code_for_Fixed-Content_Lukasiewicz_Words/">A Shift Gray Code for Fixed-Content Łukasiewicz Words</a>, Williams College, 2022.
%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.10.023">A. Combinatorial results for semigroups of order-preserving partial transformations</a>, Journal of Algebra 278, (2004), 342-359.
%H A. Laradji and A. Umar, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial results for semigroups of order-decreasing partial transformations</a>, J. Integer Seq. 7 (2004), 04.3.8.
%H Jason P. Smith, <a href="https://arxiv.org/abs/1806.01821">The poset of graphs ordered by induced containment</a>, arXiv:1806.01821 [math.CO], 2018.
%F Triangle T(n, k) read by rows; given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
%F T(n, k) = A085478(n, k)*A000108(k); A000108 = Catalan numbers. - _Philippe Deléham_, Dec 05 2003
%F Sum_{k=0..n} T(n, k)*x^k*(1-x)^(n-k) = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - _Philippe Deléham_, Aug 18 2005
%F Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - _Philippe Deléham_, Oct 18 2007
%F O.g.f. (with initial 1 excluded) is the series reversion with respect to x of (1-t*x)*x/(1+x). Cf. A062991 and A089434. - _Peter Bala_, Jul 31 2012
%F G.f.: 1 + (1 - x - T(0))/y, where T(k) = 1 - x*(1+y)/( 1 - x*y/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 03 2013
%F From _Peter Bala_, Jul 20 2015: (Start)
%F O.g.f. A(x,t) = ( 1 - x - sqrt((1 - x)^2 - 4*x*t) )/(2*x*t) = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2 + ....
%F 1 + x*(dA(x,t)/dx)/A(x,t) = 1 + (1 + t)*x + (1 + 4*t + 3*t^2)*x^2 + ... is the o.g.f. for A123160.
%F For n >= 1, the n-th row polynomial equals (1 + t)/(n+1)*Jacobi_P(n-1,1,1,2*t+1). Removing a factor of 1 + t from the row polynomials gives the row polynomials of A033282. (End)
%F From _Tom Copeland_, Jan 22 2016: (Start)
%F The o.g.f. G(x,t) = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/2x = (t + t^2) x + (t + 3t^2 + 2t^3) x^2 + (t + 6t^2 + 10t^3 + 5t^3) x^3 + ... generating shifted rows of this entry, excluding the first, was given in my 2008 formulas for A033282 with an o.g.f. f1(x,t) = G(x,t)/(1+t) for A033282. Simple transformations presented there of f1(x,t) are related to A060693 and A001263, the Narayana numbers. See also A086810.
%F The inverse of G(x,t) is essentially given in A033282 by x1, the inverse of f1(x,t): Ginv(x,t) = x [1/(t+x) - 1/(1+t+x)] = [((1+t) - t) / (t(1+t))] x - [((1+t)^2 - t^2) / (t(1+t))^2] x^2 + [((1+t)^3 - t^3) / (t(1+t))^3] x^3 - ... . The coefficients in t of Ginv(xt,t) are the o.g.f.s of the diagonals of the Pascal triangle A007318 with signed rows and an extra initial column of ones. The numerators give the row o.g.f.s of signed A074909.
%F Rows of A088617 are shifted columns of A107131, whose reversed rows are the Motzkin polynomials of A055151, related to A011973. The diagonals of A055151 give the rows of A088671, and the antidiagonals (top to bottom) of A088617 give the rows of A107131 and reversed rows of A055151. The diagonals of A107131 give the columns of A055151. The antidiagonals of A088617 (bottom to top) give the rows of A055151.
%F (End)
%F T(n, k) = [x^k] hypergeom([-n, 1 + n], [2], -x). - _Peter Luschny_, Apr 26 2022
%e Triangle begins:
%e [0] 1;
%e [1] 1, 1;
%e [2] 1, 3, 2;
%e [3] 1, 6, 10, 5;
%e [4] 1, 10, 30, 35, 14;
%e [5] 1, 15, 70, 140, 126, 42;
%e [6] 1, 21, 140, 420, 630, 462, 132;
%e [7] 1, 28, 252, 1050, 2310, 2772, 1716, 429;
%e [8] 1, 36, 420, 2310, 6930, 12012, 12012, 6435, 1430;
%e [9] 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862;
%p R := n -> simplify(hypergeom([-n, n + 1], [2], -x)):
%p Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
%p seq(print(Trow(n)), n = 0..9); # _Peter Luschny_, Apr 26 2022
%t Table[Binomial[n+k, n] Binomial[n, k]/(k+1), {n,0,10}, {k,0,n}]//Flatten (* _Michael De Vlieger_, Aug 10 2017 *)
%o (PARI) {T(n, k)= if(k+1, binomial(n+k, n)*binomial(n, k)/(k+1))}
%o (Magma) [[Binomial(n+k,n)*Binomial(n,k)/(k+1): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jun 18 2015
%o (SageMath) flatten([[binomial(n+k, 2*k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 22 2022
%Y Diagonals give A000217, A034827, A000910, A088625, A088626, A000108, A001700, A002457, A002802, A002803.
%Y Cf. A000108, A001003, A006318, A060693, A084938, A085478.
%Y Cf. A033282, A055151, A123160.
%Y Cf. A001263, A011973, A055151, A060693, A074909, A086810, A107131.
%K nonn,tabl,easy,changed
%O 0,5
%A _N. J. A. Sloane_, Nov 23 2003
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