A121747 - OEIS

%I #15 Aug 21 2024 12:26:20

%S 1,2,8,44,262,1938,15600,145086,1461888,16438446,198598860,2623055166,

%T 36933441912,560697617214,9014444990964,154698782105070,

%U 2795947673216544,53529558912435438,1074325981318055676,22703327631819203166,500976898248191064840

%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.

%H 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 Elena Barcucci, Alberto Del Lungo, and Renzo Pinzani, <a href="https://doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.

%F a(n) = Sum_{k=0..n} k * A121745(n,k).

%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);

%t d[n_] := Which[n == 1, 1, n == 2, 0, EvenQ[n], Sum[(2j - 1)!, {j, 2, n/2}], True, -d[n-1]];

%t a[n_] := a[n] = If[n == 1, 1, n*a[n-1] - d[n-1] + (n-1)!*Floor[n/2]];

%t Table[a[n], {n, 1, 22}] (* _Jean-François Alcover_, Aug 20 2024, after Maple program *)

%Y Cf. A121745, A121750.

%K nonn

%O 1,2

%A _Emeric Deutsch_, Aug 20 2006