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.
0, 0, 0, 1, 11, 101, 932, 9080, 94852, 1066644, 12905784, 167622984, 2330016768, 34551794304, 544873631616, 9110134903680, 161038110977280, 3001678242428160, 58853489050759680, 1211082030609016320, 26101332373130496000, 588033071962511616000
Offset: 0
Keywords
Examples
a(2)=0 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having all their columns starting at level zero.
Links
- Elena Barcucci, Sara Brunetti and Francesco Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
- Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
- Loïc Foissy, The antipode of of [sic] a Com-PreLie Hopf algebra, arXiv:2406.01120 [math.CO], 2024. See p. 12.
Crossrefs
Cf. A122104.
Programs
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Maple
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);
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Mathematica
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 *)
Formula
a(n) = Sum_{k>=0} k*A122104(n,k).
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.
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
E.g.f.: (2 * x + (1 - x) * log(1 - x) * (2 - log(1 - x))) / (2 * (1 - x)^2). - Ilya Gutkovskiy, Sep 02 2021
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