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A121580 Number of cells in column 1 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 A121580 #12 Sep 15 2024 07:13:47
%S A121580 1,3,11,53,317,2237,18077,164237,1656077,18348557,221561357,
%T A121580 2895986957,40737113357,613623026957,9854521894157,168083120422157,
%U A121580 3034505335078157,57810369261862157,1159018646647078157
%N A121580 Number of cells in column 1 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.
%H A121580 E. Barcucci, A. Del Lungo and R. Pinzani, <a href="https://doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.
%F A121580 a(1) = 1, a(n) = a(n-1)+(n-1)!*(1+n*(n-1)/2) for n>=2.
%F A121580 a(n) = Sum_{k=1..n} k*A100822(n,k).
%F A121580 a(n) = (1/2)*Sum_{j=0..n+1} j! - n!. - _Emeric Deutsch_, Apr 06 2008
%F A121580 Conjecture D-finite with recurrence a(n) +(-n-4)*a(n-1) +3*(n+1)*a(n-2) +2*(-2*n+3)*a(n-3) +2*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%e A121580 a(2)=3 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 2 and 1 cells in their first columns.
%p A121580 a[1]:=1: for n from 2 to 22 do a[n]:=a[n-1]+(n-1)!*(1+n*(n-1)/2) od: seq(a[n],n=1..22);
%Y A121580 Cf. A100822.
%K A121580 nonn,easy
%O A121580 1,2
%A A121580 _Emeric Deutsch_, Aug 09 2006