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Number of 2-cell columns in all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.
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%I #4 Jul 22 2022 13:26:15

%S 0,1,5,25,147,996,7668,66264,635976,6717600,77482080,969338880,

%T 13076778240,189261999360,2925629280000,48111515827200,

%U 838731380659200,15451544605593600,299960798422118400,6120505381423104000

%N Number of 2-cell columns in all deco polyominoes of height n. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column.

%C a(n)=Sum(k*A121637(n,k), k=0..n-1).

%D E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

%F a(1)=0, a(2)=1, a(3)=5, a(n)=na(n-1)+(n-1)!-(n-3)! for n>=4.

%F Conjecture D-finite with recurrence a(n) +(-2*n+1)*a(n-1) +n*(n-2)*a(n-2) +(2*n-7)*a(n-3) -(n-3)*(n-5)*a(n-4)=0. - _R. J. Mathar_, Jul 22 2022

%e a(2)=1 because the deco polyominoes of height 2 are the horizontal and vertical dominoes and only the vertical one has one 2-cell column.

%p a[1]:=0: a[2]:=1: a[3]:=5: for n from 4 to 43 do a[n]:=n*a[n-1]+(n-1)!-(n-3)! od: seq(a[n],n=1..23);

%Y Cf. A121637, A121555.

%K nonn

%O 1,3

%A _Emeric Deutsch_, Aug 14 2006