A038622 Triangular array that counts rooted polyominoes.

1, 2, 1, 5, 3, 1, 13, 9, 4, 1, 35, 26, 14, 5, 1, 96, 75, 45, 20, 6, 1, 267, 216, 140, 71, 27, 7, 1, 750, 623, 427, 238, 105, 35, 8, 1, 2123, 1800, 1288, 770, 378, 148, 44, 9, 1, 6046, 5211, 3858, 2436, 1296, 570, 201, 54, 10, 1, 17303, 15115, 11505, 7590, 4302, 2067, 825, 265

Offset: 0

Author

N. J. A. Sloane, torsten.sillke(AT)lhsystems.com

Keywords

• nonn
• tabl
• easy
• nice

Comments

The PARI program gives any row k and any n-th term for this triangular array in square or right triangle array format. - Randall L Rathbun, Jan 20 2002

Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

Triangle read by rows = partial sums of A064189 terms starting from the right. - Gary W. Adamson, Oct 25 2008

Column k has e.g.f. exp(x)*(Bessel_I(k,2x)+Bessel_I(k+1,2x)). - Paul Barry, Mar 08 2011

Examples

			From _Paul Barry_, Mar 08 2011: (Start)
Triangle begins
     1;
     2,    1;
     5,    3,    1;
    13,    9,    4,   1;
    35,   26,   14,   5,   1;
    96,   75,   45,  20,   6,   1;
   267,  216,  140,  71,  27,   7,  1;
   750,  623,  427, 238, 105,  35,  8, 1;
  2123, 1800, 1288, 770, 378, 148, 44, 9, 1;
Production matrix is
  2, 1,
  1, 1, 1,
  0, 1, 1, 1,
  0, 0, 1, 1, 1,
  0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 1, 1
(End)
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357. See Table 1 on page 340.

Crossrefs

Cf. A005773 (1st column), A005774 (2nd column), A005775, A066822, A000244 (row sums).

Cf. A064189, A007318, A061554, A092392.

Programs

Haskell

import Data.List (transpose)
a038622 n k = a038622_tabl !! n !! k
a038622_row n = a038622_tabl !! n
a038622_tabl = iterate (\row -> map sum $
   transpose [tail row ++ [0,0], row ++ [0], [head row] ++ row]) [1]
-- Reinhard Zumkeller, Feb 26 2013

Maple

T := (n,k) -> simplify(GegenbauerC(n-k,-n+1,-1/2)+GegenbauerC(n-k-1,-n+1,-1/2)):
for n from 1 to 9 do seq(T(n,k),k=1..n) od; # Peter Luschny, May 12 2016

Mathematica

nmax = 10; t[n_ /; n > 0, k_ /; k >= 1] := t[n, k] = t[n-1, k-1] + t[n-1, k] + t[n-1, k+1]; t[0, 0] = 1; t[0, ] = 0; t[?Negative, ?Negative] = 0; t[n, 0] := 2 t[n-1, 0] + t[n-1, 1]; Flatten[ Table[ t[n, k], {n, 0, nmax}, {k, 0, n}]](* Jean-François Alcover, Nov 09 2011 *)

PARI

s=[0,1]; {A038622(n,k)=if(n==0,1,t=(2*(n+k)*(n+k-1)*s[2]+3*(n+k-1)*(n+k-2)*s[1])/((n+2*k-1)*n); s[1]=s[2]; s[2]=t; t)}

Formula

a(n, k) = a(n-1, k-1) + a(n-1, k) + a(n-1, k+1) for k>0, a(n, k) = 2*a(n-1, k) + a(n-1, k+1) for k=0.

Riordan array ((sqrt(1-2x-3x^2)+3x-1)/(2x(1-3x)),(1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array ((1-x)/(1+x+x^2),x/(1+x+x^2)). First column is A005773(n+1). Row sums are 3^n (A000244). If L=A038622, then L*L' is the Hankel matrix for A005773(n+1), where L' is the transpose of L. - Paul Barry, Sep 18 2006

T(n,k) = GegenbauerC(n-k,-n+1,-1/2) + GegenbauerC(n-k-1,-n+1,-1/2). In this form also the missing first column of the triangle 1,1,1,3,7,19,... (cf. A002426) can be computed. - Peter Luschny, May 12 2016

From Peter Bala, Jul 12 2021: (Start)

T(n,k) = Sum_{j = k..n} binomial(n,j)*binomial(j,floor((j-k)/2)).

Matrix product of Riordan arrays ( 1/(1 - x), x/(1 - x) ) * ( (1 - x*c(x^2))/(1 - 2*x), x*c(x^2) ) = A007318 * A061554 (triangle version), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

Triangle equals A007318^(-1) * A092392 * A007318. (End)

The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + x)*(1 + x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

Extensions

More terms from David W. Wilson