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A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282. 23
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, 132, 0, 1, 27, 225, 825, 1485, 1287, 429, 0, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 0, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 0, 1, 54, 936, 7644, 34498, 91728 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Mirror image of triangle A133336. - Philippe Deléham, Dec 10 2008

From Tom Copeland, Oct 09 2011: (Start)

With polynomials

P(0,t) = 0

P(1,t) = 1

P(2,t) = t

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

P(4,t) = t + 5 t^2 +  5 t^3

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

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

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

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)

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

Diagonals of A132081 are essentially rows of this sequence. - Tom Copeland, May 08 2012

T(r, s) gives the number of [0,r]-covering hierarchies with s segments (see Kreweras). - Michel Marcus, Nov 22 2014

LINKS

Table of n, a(n) for n=0..61.

Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.

Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.

V. Buchstaber and E. Bunkova,Elliptic formal group laws, integral Hirzebruch genera and Kirchever genera,, arXiv:1010.0944 [math-ph], 2010 (see p. 19).

V. Buchstaber and T. Panov, Toric Topology. Chapter 1: Geometry and Combinatorics of Polytopes,, arXiv:1102.1079 [math.CO], 2011-2012 (see p. 41).

G. Chatel, V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014, 2015.

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.

T. Copeland, Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra,, 2014.

T. Copeland, Lagrange a la Lah, 2011.

B. Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7

G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973, p. 21-22.

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

FORMULA

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.

For k>0, T(n, k) = binomial(n+k-1, n)*binomial(n+2k, k)/(n+k+1); T(0, 0) = 1 and T(n, 0) = 0 if n > 0.

Sum_{k>=0} T(n, k)*2^k = A107841(n). - Philippe Deléham, May 26 2005

Sum_{k>=0} T(n-k, k) = A005043(n). - Philippe Deléham, May 30 2005

T(n, k) = A108263(n+k, k). - Philippe Deléham, May 30 2005

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

Sum_{k=0..n} T(n,k)*5^k*(-2)^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008

Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A154825(n). - Philippe Deléham, Jan 17 2009

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

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

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

EXAMPLE

Triangle starts:

  1;

  0,  1;

  0,  1,  2;

  0,  1,  5,  5;

  0,  1,  9, 21, 14;

  ...

MATHEMATICA

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 *)

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 *)

PROG

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

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 22 2014

CROSSREFS

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

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

Row sums: A001003 (Schroeder numbers).

Cf. A033282, A084938.

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

Sequence in context: A085650 A201910 A109450 * A085838 A094456 A010028

Adjacent sequences:  A086807 A086808 A086809 * A086811 A086812 A086813

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Aug 05 2003

STATUS

approved

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Last modified October 20 18:01 EDT 2018. Contains 316399 sequences. (Running on oeis4.)