OFFSET
0,5
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.
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
EXAMPLE
a(2)=0 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having all their columns starting at level zero.
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);
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Aug 24 2006
STATUS
approved