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A121582 Number of cells in column 2 of 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.

%I A121582 #9 Jul 26 2022 14:05:07
%S A121582 0,1,7,40,252,1837,15259,141798,1455694,16360387,199845957,2637020884,
%T A121582 37388864368,566971338009,9157693715407,156975522127762,
%U A121582 2846305448882274,54432896145210943,1095019542858729769
%N A121582 Number of cells in column 2 of 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 A121582 a(n)=Sum(k*A121581(n,k),k=0..n-1).
%D A121582 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%H A121582 Harvey P. Dale, <a href="/A121582/b121582.txt">Table of n, a(n) for n = 1..400</a>
%F A121582 a(1)=0, a(2)=1, a(n)=[(2n-3)a(n-1)-(n-1)a(n-2)+(n-1)!(n-2)(n^2-3n+4)/2]/(n-2) for n>=3.
%F A121582 a(n) ~ n*n!/2. - _Vaclav Kotesovec_, Aug 15 2013
%F A121582 D-finite with recurrence (-49*n+454)*a(n) +(49*n^2-454*n-1328)*a(n-1) +(49*n^2+1553*n-1464)*a(n-2) +(-581*n^2+612*n+1035)*a(n-3) +(819*n^2-3682*n+4007)*a(n-4) -4*(84*n-169)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jul 26 2022
%e A121582 a(2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 1 cells in their second columns.
%p A121582 a:=proc(n) if n=1 then 0 elif n=2 then 1 else ((2*n-3)*a(n-1)-(n-1)*a(n-2)+(n-1)!*(n-2)*(n^2-3*n+4)/2)/(n-2) fi end: seq(a(n),n=1..22);
%t A121582 RecurrenceTable[{a[1]==0,a[2]==1,a[n]==((2n-3)a[n-1]-(n-1)a[n-2]+ (n-1)!(n-2) (n^2-3n+4)/2)/(n-2)},a,{n,20}] (* _Harvey P. Dale_, Oct 23 2012 *)
%Y A121582 Cf. A121580, A121581, A121584.
%K A121582 nonn
%O A121582 1,3
%A A121582 _Emeric Deutsch_, Aug 11 2006