|
%I #4 Jul 26 2022 11:21:05
%S 1,2,8,44,262,1938,15600,145086,1461888,16438446,198598860,2623055166,
%T 36933441912,560697617214,9014444990964,154698782105070,
%U 2795947673216544,53529558912435438,1074325981318055676
%N Number of columns of odd length 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.
%C a(n)=Sum(k*A121745(n,k), k=0..n).
%D E. Barcucci, S. Brunetti and F. Del Ristoro, Succession rules and deco polyominoes, Theoret. Informatics Appl., 34, 2000, 1-14.
%D E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%F Recurrence relation: a(n)=n*a[n-1]-d(n-1)+(n-1)!*floor(n/2) for n>=2, a(1)=1, where d(1)=1, d(2)=0, d(2n)=3!+5!+...+(2n-1)!, d(2n+1)=-d(2n).
%F Conjecture D-finite with recurrence (-9*n+38)*a(n) +3*(3*n^2-19*n+27)*a(n-1) +(9*n^3-37*n^2-24*n+127)*a(n-2) +(-9*n^4+102*n^3-392*n^2+525*n-85)*a(n-3) -(n-3)*(28*n^3-223*n^2+510*n-237)*a(n-4) -(19*n-15)*(n-3)*(n-4)^2*a(n-5)=0. - _R. J. Mathar_, Jul 26 2022
%e a(2)=2 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having 0 and 2 columns of odd length, respectively.
%p d:=proc(n) if n=1 then 1 elif n=2 then 0 elif n mod 2 = 0 then add((2*j-1)!,j=2..n/2) else -d(n-1) fi end: a[1]:=1: for n from 2 to 22 do a[n]:=n*a[n-1]-d(n-1)+(n-1)!*floor(n/2) od: seq(a[n],n=1..22);
%Y Cf. A121745, A121750.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Aug 20 2006
|
