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

%I A066822 #32 Jan 08 2025 09:26:19
%S A066822 1,5,20,71,238,770,2436,7590,23397,71566,217646,659022,1988805,
%T A066822 5986176,17980968,53922096,161492571,483149385,1444245936,4314214443,
%U A066822 12880107548,38436170366,114657076900,341926185770,1019435748435,3038815305981,9056974493700
%N A066822 The fourth column of A038622, triangular array that counts rooted polyominoes.
%C A066822 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).
%H A066822 Reinhard Zumkeller, <a href="/A066822/b066822.txt">Table of n, a(n) for n = 0..1000</a>
%H A066822 D. Gouyou-Beauchamps and G. Viennot, <a href="http://dx.doi.org/10.1016/0196-8858(88)90017-6">Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem</a>, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.
%F A066822 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).
%F A066822 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
%F A066822 a(n) ~ 3^(n+7/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 10 2014
%F A066822 a(n) = GegenbauerC(n,-n+1-4,-1/2)+GegenbauerC(n-1,-n-3,-1/2). - _Peter Luschny_, May 12 2016
%p A066822 a := n -> simplify(GegenbauerC(n,-n+1-4,-1/2)+GegenbauerC(n-1,-n-3,-1/2)):
%p A066822 seq(a(n), n=0..20); # _Peter Luschny_, May 12 2016
%t A066822 Table[GegenbauerC[n,-n-3,-1/2]+GegenbauerC[n-1,-n-3,-1/2],{n,0,40}] (* _Harvey P. Dale_, Feb 20 2017 *)
%o A066822 (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)}
%o A066822 (Haskell)
%o A066822 a066822 = flip a038622 3 . (+ 3)  -- _Reinhard Zumkeller_, Feb 26 2013
%Y A066822 Cf. A038622.
%Y A066822 Cf. A005773, A005774, A005775.
%K A066822 easy,nice,nonn
%O A066822 0,2
%A A066822 _Randall L Rathbun_, Jan 19 2002
%E A066822 More terms from _Harvey P. Dale_, Feb 20 2017