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A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

%I A086810 #125 May 05 2023 01:37:07
%S A086810 1,0,1,0,1,2,0,1,5,5,0,1,9,21,14,0,1,14,56,84,42,0,1,20,120,300,330,
%T A086810 132,0,1,27,225,825,1485,1287,429,0,1,35,385,1925,5005,7007,5005,1430,
%U A086810 0,1,44,616,4004,14014,28028,32032,19448,4862,0,1,54,936,7644,34398,91728
%N A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.
%C A086810 Mirror image of triangle A133336. - _Philippe Deléham_, Dec 10 2008
%C A086810 From _Tom Copeland_, Oct 09 2011: (Start)
%C A086810 With polynomials
%C A086810 P(0,t) = 0
%C A086810 P(1,t) = 1
%C A086810 P(2,t) = t
%C A086810 P(3,t) = t + 2 t^2
%C A086810 P(4,t) = t + 5 t^2 +  5 t^3
%C A086810 P(5,t) = t + 9 t^2 + 21 t^3 + 14 t^4
%C A086810 The o.g.f. A(x,t) = {1+x-sqrt[(1-x)^2-4xt]}/[2(1+t)] (see Drake et al.).
%C A086810 B(x,t)= x-t x^2/(1-x)= x-t(x^2+x^3+x^4+...) is the comp. inverse in x.
%C A086810 Let h(x,t) = 1/(dB/dx) = (1-x)^2/(1+(1+t)*x*(x-2)) = 1/(1-t(2x+3x^2+4x^3+...)), an o.g.f. in x for row polynomials in t of A181289. Then P(n,t) is given by (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A = exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). These results are a special case of A133437 with u(x,t) = B(x,t), i.e., u_1=1 and (u_n)=-t for n > 1. See A001003 for t = 1. (End)
%C A086810 Let U(x,t) = [A(x,t)-x]/t, then U(x,0) = -dB(x,t)/dt and U satisfies dU/dt = UdU/dx, the inviscid Burgers' equation (Wikipedia), also called the Hopf equation (see Buchstaber et al.). Also U(x,t) = U(A(x,t),0) = U(x+tU,0) since U(x,0) = [x-B(x,t)]/t. - _Tom Copeland_, Mar 12 2012
%C A086810 Diagonals of A132081 are essentially rows of this sequence. - _Tom Copeland_, May 08 2012
%C A086810 T(r, s) is the number of [0,r]-covering hierarchies with s segments (see Kreweras). - _Michel Marcus_, Nov 22 2014
%C A086810 From _Yu Hin Au_, Dec 07 2019: (Start)
%C A086810 T(n,k) is the number of small Schröder n-paths (lattice paths from (0,0) to (2n,0) using steps U=(1,1), F=(2,0), D=(1,-1) with no F step on the x-axis) that has exactly k U steps.
%C A086810 T(n,k) is the number of Schröder trees (plane rooted tree where each internal node has at least two children) with exactly n+1 leaves and k internal nodes. (End)
%H A086810 Michael De Vlieger, <a href="/A086810/b086810.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened.)
%H A086810 Yu Hin (Gary) Au, <a href="https://arxiv.org/abs/1912.00555">Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers</a>, arXiv:1912.00555 [math.CO], 2019.
%H A086810 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry3/barry252.html">On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.6.
%H A086810 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H A086810 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 A086810 V. Buchstaber and E. Bunkova,<a href="http://arxiv.org/abs/1010.0944">Elliptic formal group laws, integral Hirzebruch genera and Kirchever genera,</a>, arXiv:1010.0944 [math-ph], 2010 (see p. 19).
%H A086810 V. Buchstaber and T. Panov,<a href="http://arxiv.org/abs/1102.1079"> Toric Topology. Chapter 1: Geometry and Combinatorics of Polytopes,</a>, arXiv:1102.1079 [math.CO], 2011-2012 (see p. 41).
%H A086810 G. Chatel, V. Pilaud, <a href="http://arxiv.org/abs/1411.3704">Cambrian Hopf Algebras</a>, arXiv:1411.3704 [math.CO], 2014-2015.
%H A086810 T. Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015.
%H A086810 T. Copeland, <a href="https://tcjpn.wordpress.com/2014/09/17/compositional-inverse-pairs-the-inviscid-burgers-hopf-equation-and-the-stasheff-associahedra/"> Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra</a>, 2014.
%H A086810 T. Copeland, <a href="https://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>, 2011.
%H A086810 B. Drake, Ira M. Gessel, and Guoce Xin, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Gessel/gessel20.html">Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry,</a> J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
%H A086810 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973, p. 21-22.
%H A086810 G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
%H A086810 J. Zhou, <a href="http://arxiv.org/abs/1405.5296">Quantum deformation theory of the Airy curve and the mirror symmetry of a point</a>, arXiv preprint arXiv:1405.5296 [math.AG], 2014.
%F A086810 Triangle T(n, k) read by rows; given by [0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
%F A086810 For k>0, T(n, k) = binomial(n-1, k-1)*binomial(n+k, k)/(n+1); T(0, 0) = 1 and T(n, 0) = 0 if n > 0. [corrected by _Marko Riedel_, May 04 2023]
%F A086810 Sum_{k>=0} T(n, k)*2^k = A107841(n). - _Philippe Deléham_, May 26 2005
%F A086810 Sum_{k>=0} T(n-k, k) = A005043(n). - _Philippe Deléham_, May 30 2005
%F A086810 T(n, k) = A108263(n+k, k). - _Philippe Deléham_, May 30 2005
%F A086810 Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - _Philippe Deléham_, Nov 05 2007
%F A086810 Sum_{k=0..n} T(n,k)*5^k*(-2)^(n-k) = A152601(n). - _Philippe Deléham_, Dec 10 2008
%F A086810 Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A154825(n). - _Philippe Deléham_, Jan 17 2009
%F A086810 Umbrally, P(n,t) = Lah[n-1,-t*a.]/n! = (1/n)*Sum_{k=1..n-1} binomial(n-2,k-1)a_k t^k/k!, where (a.)^k = a_k = (n-1+k)!/(n-1)!, the rising factorial, and Lah(n,t) = n!*Laguerre(n,-1,t) are the Lah polynomials A008297 related to the Laguerre polynomials of order -1. - _Tom Copeland_, Oct 04 2014
%F A086810 T(n, k) = binomial(n, k)*binomial(n+k, k-1)/n, for k >= 0; T(0, 0) = 1 (see Kreweras, p. 21). - _Michel Marcus_, Nov 22 2014
%F A086810 P(n,t) = Lah[n-1,-:Dt:]/n! t^(n-1) with (:Dt:)^k = (d/dt)^k t^k = k! Laguerre(k,0,-:tD:) with (:tD:)^j = t^j D^j. The normalized Laguerre polynomials of 0 order are given in A021009. - _Tom Copeland_, Aug 22 2016
%e A086810 Triangle starts:
%e A086810   1;
%e A086810   0,  1;
%e A086810   0,  1,  2;
%e A086810   0,  1,  5,  5;
%e A086810   0,  1,  9, 21, 14;
%e A086810   ...
%t A086810 Table[Boole[n == 2] + If[# == -1, 0, Binomial[n - 3, #] Binomial[n + # - 1, #]/(# + 1)] &[k - 1], {n, 2, 12}, {k, 0, n - 2}] // Flatten (* after _Jean-François Alcover_ at A033282, or *)
%t A086810 Table[If[n == 0, 1, Binomial[n, k] Binomial[n + k, k - 1]/n], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Aug 22 2016 *)
%o A086810 (PARI) t(n, k) = if (n==0, 1, binomial(n, k)*binomial(n+k, k-1)/n);
%o A086810 tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n,k), ", ");); print(););} \\ _Michel Marcus_, Nov 22 2014
%Y A086810 Diagonals: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281.
%Y A086810 The diagonals (except for A000007) are also the diagonals of A033282.
%Y A086810 Row sums: A001003 (Schroeder numbers).
%Y A086810 Cf. A033282, A084938.
%Y A086810 Cf. A001003, A008297, A021009, A132081, A133437, A181289.
%K A086810 easy,nonn,tabl
%O A086810 0,6
%A A086810 _Philippe Deléham_, Aug 05 2003
%E A086810 Typo in a(60) corrected by _Michael De Vlieger_, Nov 21 2019