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A121555 Number of 1-cell columns in all deco polyominoes of height n.

%I A121555 #26 May 07 2025 09:13:03
%S A121555 1,2,7,32,178,1164,8748,74304,704016,7362720,84255840,1047358080,
%T A121555 14054739840,202514376960,3118666924800,51119166873600,
%U A121555 888640952371200,16330301780889600,316322420114534400,6441691128993792000,137586770616637440000,3075566993729556480000
%N A121555 Number of 1-cell columns in all deco polyominoes of height n.
%C A121555 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 A121555 It appears that a(n) is a function of the harmonic numbers. [_Gary Detlefs_, Aug 13 2010]
%H A121555 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.
%H A121555 Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.7.8.
%F A121555 a(n) = Sum_{k=0..n} k*A121554(n, k).
%F A121555 a(1) = 1, a(n) = n*a(n-1)+(n-2)!*(n-2) for n >= 2.
%F A121555 a(n) = n!*(h(n) - (n-1)/n), where h(n) = Sum_{k=1..n} 1/k. - _Gary Detlefs_, Aug 13 2010
%F A121555 (-n+3)*a(n) + (2*n^2-7*n+4)*a(n-1) - (n-1)*(n-2)^2*a(n-2) = 0. - _R. J. Mathar_, Jul 15 2017
%F A121555 a(n) = abs(Stirling1(n + 1, 2)) - (n - 1)*(n - 1)!. - _Detlef Meya_, Apr 09 2024
%e A121555 a(2)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 0 and 2 columns with exactly 1 cell.
%p A121555 a[1]:=1: for n from 2 to 23 do a[n]:=n*a[n-1]+(n-2)!*(n-2) od:
%p A121555 seq(a[n], n = 1..23);
%p A121555 # Alternative:
%p A121555 a := n -> (n - 1)! * (n*harmonic(n) - (n - 1)):
%p A121555 seq(a(n), n = 1..22);  # _Peter Luschny_, Apr 09 2024
%t A121555 a[n_]:=Abs[StirlingS1[n+1,2]]-(n-1)*(n-1)!;Flatten[Table[a[n],{n,1,22}]] (* _Detlef Meya_, Apr 09 2024 *)
%Y A121555 Cf. A121554.
%K A121555 nonn
%O A121555 1,2
%A A121555 _Emeric Deutsch_, Aug 08 2006