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A038622 Triangular array that counts rooted polyominoes. 29
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

D. Gouyou-Beauchamps, 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.

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

EXAMPLE

1; 2,1; 5,3,1; 13,9,4,1; ...

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

- Paul Barry, Mar 08 2011

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 *)

PROG

(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)}

(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

CROSSREFS

Cf. A005773, A005774, A005775, A066822.

CF. A064189. - Gary W. Adamson, Oct 25 2008

Sequence in context: A047858 A125171 A048472 * A193954 A162997 A112339

Adjacent sequences:  A038619 A038620 A038621 * A038623 A038624 A038625

KEYWORD

tabl,easy,nice,nonn

AUTHOR

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

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified December 11 08:51 EST 2016. Contains 279052 sequences.