A086810 - OEIS

%I #125 May 05 2023 01:37:07

%S 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 132,0,1,27,225,825,1485,1287,429,0,1,35,385,1925,5005,7007,5005,1430,

%U 0,1,44,616,4004,14014,28028,32032,19448,4862,0,1,54,936,7644,34398,91728

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

%C Mirror image of triangle A133336. - _Philippe Deléham_, Dec 10 2008

%C From _Tom Copeland_, Oct 09 2011: (Start)

%C With polynomials

%C P(0,t) = 0

%C P(1,t) = 1

%C P(2,t) = t

%C P(3,t) = t + 2 t^2

%C P(4,t) = t + 5 t^2 +  5 t^3

%C P(5,t) = t + 9 t^2 + 21 t^3 + 14 t^4

%C The o.g.f. A(x,t) = {1+x-sqrt[(1-x)^2-4xt]}/[2(1+t)] (see Drake et al.).

%C B(x,t)= x-t x^2/(1-x)= x-t(x^2+x^3+x^4+...) is the comp. inverse in x.

%C 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 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 Diagonals of A132081 are essentially rows of this sequence. - _Tom Copeland_, May 08 2012

%C T(r, s) is the number of [0,r]-covering hierarchies with s segments (see Kreweras). - _Michel Marcus_, Nov 22 2014

%C From _Yu Hin Au_, Dec 07 2019: (Start)

%C 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 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 Michael De Vlieger, <a href="/A086810/b086810.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened.)

%H 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 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 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 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 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 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 G. Chatel, V. Pilaud, <a href="http://arxiv.org/abs/1411.3704">Cambrian Hopf Algebras</a>, arXiv:1411.3704 [math.CO], 2014-2015.

%H 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 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 T. Copeland, <a href="https://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>, 2011.

%H 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 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 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 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 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 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 Sum_{k>=0} T(n, k)*2^k = A107841(n). - _Philippe Deléham_, May 26 2005

%F Sum_{k>=0} T(n-k, k) = A005043(n). - _Philippe Deléham_, May 30 2005

%F T(n, k) = A108263(n+k, k). - _Philippe Deléham_, May 30 2005

%F 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 Sum_{k=0..n} T(n,k)*5^k*(-2)^(n-k) = A152601(n). - _Philippe Deléham_, Dec 10 2008

%F Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A154825(n). - _Philippe Deléham_, Jan 17 2009

%F 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 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 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 Triangle starts:

%e   1;

%e   0,  1;

%e   0,  1,  2;

%e   0,  1,  5,  5;

%e   0,  1,  9, 21, 14;

%e   ...

%t 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 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 (PARI) t(n, k) = if (n==0, 1, binomial(n, k)*binomial(n+k, k-1)/n);

%o tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n,k), ", ");); print(););} \\ _Michel Marcus_, Nov 22 2014

%Y Diagonals: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281.

%Y The diagonals (except for A000007) are also the diagonals of A033282.

%Y Row sums: A001003 (Schroeder numbers).

%Y Cf. A033282, A084938.

%Y Cf. A001003, A008297, A021009, A132081, A133437, A181289.

%K easy,nonn,tabl

%O 0,6

%A _Philippe Deléham_, Aug 05 2003

%E Typo in a(60) corrected by _Michael De Vlieger_, Nov 21 2019