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 A066822 The fourth row of A038622, triangular array that counts rooted polyominoes. 4
 1, 5, 20, 71, 238, 770, 2436, 7590, 23397, 71566, 217646, 659022, 1988805, 5986176, 17980968, 53922096, 161492571, 483149385, 1444245936, 4314214443, 12880107548 (list; graph; refs; listen; history; text; internal format)
 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) REFERENCES 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. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 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 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: A054444 A121332 A122695 * A137212 A118049 A114247 Adjacent sequences:  A066819 A066820 A066821 * A066823 A066824 A066825 KEYWORD easy,nice,nonn AUTHOR Randall L. Rathbun, Jan 19 2002 STATUS approved

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