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A122105
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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.
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2
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0, 0, 0, 1, 11, 101, 932, 9080, 94852, 1066644, 12905784, 167622984, 2330016768, 34551794304, 544873631616, 9110134903680, 161038110977280, 3001678242428160, 58853489050759680, 1211082030609016320, 26101332373130496000, 588033071962511616000
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OFFSET
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0,5
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LINKS
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FORMULA
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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
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EXAMPLE
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a(2)=0 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having all their columns starting at level zero.
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MAPLE
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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
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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