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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).
0
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
1,3
COMMENTS
Row n contains n(n+1)/2 terms. Row sums yield A007808. Column sums yield the odd-indexed Fibonacci numbers (A001519).
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.
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)]
EXAMPLE
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;
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;
CROSSREFS
Sequence in context: A094449 A274776 A274777 * A278213 A034093 A246187
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jan 21 2008
STATUS
approved