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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

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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 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.

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