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A122105 Sum of the bottom levels of all 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.

%I A122105 #25 Jun 06 2024 12:32:44
%S A122105 0,0,0,1,11,101,932,9080,94852,1066644,12905784,167622984,2330016768,
%T A122105 34551794304,544873631616,9110134903680,161038110977280,
%U A122105 3001678242428160,58853489050759680,1211082030609016320,26101332373130496000,588033071962511616000
%N A122105 Sum of the bottom levels of all 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.
%H A122105 Elena Barcucci, Sara Brunetti and Francesco Del Ristoro, <a href="http://www.numdam.org/item?id=ITA_2000__34_1_1_0">Succession rules and deco polyominoes</a>, Theoret. Informatics Appl., 34, 2000, 1-14.
%H A122105 Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.
%H A122105 Loïc Foissy, <a href="https://arxiv.org/abs/2406.01120">The antipode of of [sic] a Com-PreLie Hopf algebra</a>, arXiv:2406.01120 [math.CO], 2024. See p. 12.
%F A122105 a(n) = Sum_{k>=0} k*A122104(n,k).
%F A122105 Recurrence relation: a(n) = (2n-1)*a(n-1)-(n-1)^2*a(n-2)+(n-2)!*(n-2) for n>=3, a(0)=a(1)=a(2)=0.
%F A122105 a(n) = n![n - H(n) - (H(n))^2/2 + (1/2)Sum(1/j^2, j=1..n)], where H(n)=Sum(1/j, j=1..n). - _Emeric Deutsch_, Apr 06 2008
%F A122105 E.g.f.: (2 * x + (1 - x) * log(1 - x) * (2 - log(1 - x))) / (2 * (1 - x)^2). - _Ilya Gutkovskiy_, Sep 02 2021
%F A122105 D-finite with recurrence a(n) +(-3*n+1)*a(n-1) +(3*n^2-4*n-2)*a(n-2) +(-n^3+2*n^2+7*n-15)*a(n-3) +(n-3)^3*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%e A122105 a(2)=0 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having all their columns starting at level zero.
%p A122105 a[0]:=0: a[1]:=0: a[2]:=0: for n from 3 to 23 do a[n]:=(2*n-1)*a[n-1]-(n-1)^2*a[n-2]+(n-2)*(n-2)! od: seq(a[n],n=0..23);
%t A122105 RecurrenceTable[{a[0]==a[1]==0,a[n]==(2n-1)*a[n-1]-(n-1)^2*a[n-2]+(n-2)!*(n-2)}, a, {n,0,20}] (* _Harvey P. Dale_, Dec 04 2014; adapted to offset 0 by _Georg Fischer_, Jul 30 2022 *)
%Y A122105 Cf. A122104.
%K A122105 nonn
%O A122105 0,5
%A A122105 _Emeric Deutsch_, Aug 24 2006