A136129 Triangle read by rows: T(n,k) is the number of directed, vertically convex polyominoes of height n and area k (n<= k <=n(n+1)/2).

1, 0, 2, 1, 0, 0, 4, 5, 3, 1, 0, 0, 0, 8, 15, 17, 15, 9, 4, 1, 0, 0, 0, 0, 16, 39, 59, 75, 78, 67, 48, 29, 14, 5, 1, 0, 0, 0, 0, 0, 32, 95, 175, 269, 358, 419, 432, 400, 334, 250, 166, 97, 49, 20, 6, 1, 0, 0, 0, 0, 0, 0, 64, 223, 479, 845, 1300, 1801, 2269, 2622, 2805, 2794, 2593

Offset: 1

Author

Emeric Deutsch, Jan 21 2008

Keywords

• nonn
• tabf

Comments

Row n contains n(n+1)/2 terms. Row sums yield A007808. Column sums yield the odd-indexed Fibonacci numbers (A001519).

Examples

			Triangle starts:
1;
0,2,1;
0,0,4,5,3,1;
0,0,0,8,15,17,15,9,4,1;
0,0,0,0,16,39,59,75,78,67,48,29,14,5,1;
		

Links

• E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos dirigés verticalement convexes, Séminaire Lotharingien de Combinatoire, B31d (1993), 11 pp.

Crossrefs

Cf. A007808, A001519.

Programs

Maple

A:=t*z*(1-t)/(1-t-2*t*z+t^2*z): B:=t^2*z*(z-1)/((1-t-2*t*z+t^2*z)*(1-t*z)): Aser:=simplify(series(A,z=0,12)): Bser:=simplify(series(B,z=0,12)): for n to 12 do A[n]:=coeff(Aser,z,n): B[n]:=coeff(Bser,z,n) end do: P[1]:=A[1]: for n from 2 to 7 do P[n]:=sort(expand(simplify(A[n]+add(B[n-j]*P[j]*t^j,j=1..n-1)))) end do: for n to 7 do seq(coeff(P[n],t,j),j=1..(1/2)*n*(n+1)) end do;

Formula

G.f. G(t,z) satisfies G(t,z)=zt(1-t)/(1-t-2zt+zt^2) +z(z-1)t^2*G(t,tz)/[(1-t-2zt+zt^2)(1-zt)]