A066822 The fourth column of A038622, triangular array that counts rooted polyominoes.

1, 5, 20, 71, 238, 770, 2436, 7590, 23397, 71566, 217646, 659022, 1988805, 5986176, 17980968, 53922096, 161492571, 483149385, 1444245936, 4314214443, 12880107548, 38436170366, 114657076900, 341926185770, 1019435748435, 3038815305981, 9056974493700

Offset: 0

Author

Randall L Rathbun, Jan 19 2002

Keywords

• easy
• nice
• nonn

Comments

There is a general solution for all rows of this triangular array: For the k-th row and n-th term on this row: a(0)=0; a(1)=1; a(n) = (2*k-1+n)*n*a(n) = 2*(n+k)*(n+k-1)*a(n-1) + 3*(n+k-1)*(n+k-2)*a(n-2).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• 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.

Crossrefs

Cf. A038622.

Cf. A005773, A005774, A005775.

Programs

Haskell

a066822 = flip a038622 3 . (+ 3)  -- Reinhard Zumkeller, Feb 26 2013

Maple

a := n -> simplify(GegenbauerC(n,-n+1-4,-1/2)+GegenbauerC(n-1,-n-3,-1/2)):
seq(a(n), n=0..20); # Peter Luschny, May 12 2016

Mathematica

Table[GegenbauerC[n,-n-3,-1/2]+GegenbauerC[n-1,-n-3,-1/2],{n,0,40}] (* Harvey P. Dale, Feb 20 2017 *)

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(0)=0; a(1)=1; (n+7)*n*a(n)=2*(n+4)*(n+3)*a(n-1) + 3*(n+3)*(n+2)*a(n-2).

a(n) = ((-3)^(1/2)/9)*(-2*(n+7)^(-1)*(n+4)*(-1)^n*hypergeom([3/2, n+6],[2],4/3)-(n+6)^(-1)*(-1)^n*(5*n+18)*hypergeom([3/2, n+5],[2],4/3)). - Mark van Hoeij, Oct 31 2011

a(n) ~ 3^(n+7/2) / sqrt(Pi*n). - Vaclav Kotesovec, Mar 10 2014

a(n) = GegenbauerC(n,-n+1-4,-1/2)+GegenbauerC(n-1,-n-3,-1/2). - Peter Luschny, May 12 2016

Extensions

More terms from Harvey P. Dale, Feb 20 2017