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

0,2

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, 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.

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

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 *)

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

Crossrefs

Cf. A038622.

Cf. A005773, A005774, A005775.

Sequence in context: A121332 A122695 A269914 * A137212 A346398 A270080

Adjacent sequences: A066819 A066820 A066821 * A066823 A066824 A066825

Keyword

easy,nice,nonn

Author

Randall L Rathbun, Jan 19 2002

Extensions

More terms from Harvey P. Dale, Feb 20 2017

Status

approved