0,5

The originator sequences are A142963 and A156919.

The Flower Triangle seems to be an appropriate name for the triangular array of this sequence. The zero patterns of the Flower Polynomials of the first, see A156921, the second, see A156925, the third, see A156927, and the fourth kind, see A156933, look like flowers.

The first Maple program generates the Flower Triangle sequence.

The second program generates the Right Hand Columns sequences and the third one generates the Left Hand Column sequences. For an explanation of these two algorithms see sequence A142963 by Wolfdieter Lang.

FT(n,m) = (m+1)*FT(n-1,m) + (2n-2m+1)*FT(n-1,m-1) with FT(n,m=0)=1 and FT(n,n)=1. In view of the offset n=0,1,2,... and m=0,1,..,n.

The first few rows of the triangle are:

[1]

[1, 1]

[1, 5 , 1 ]

[1, 15, 18, 1]

[1, 37, 129, 58, 1]

[1, 83, 646, 877, 179, 1]

nmax:=8: mmax:=nmax: for n from 0 to nmax do A[n, 0]:=1: end do: for m from 1 to mmax do A[0, m]:=0 end do: for n from 1 to nmax do for m from 1 to mmax do A[n, m]:=(m+1)*A[n-1, m] + (2*n-2*m+1) * A[n-1, m-1] end do: end do: for n from 0 to nmax do for m from 0 to n do FT[n*(n+1)/2+m]:=A[n, m] end do end do: a:=n-> FT[n]: seq(a(n), n=0..((1/2)*nmax^2+(3/2)*nmax));

RHCnr:=5; RHCmax:=10; RHCend:=RHCnr+RHCmax: for k from RHCnr to RHCend do for n from 0 to k do S2[k, n]:=sum((-1)^(n+i)*binomial(n, i)*i^k/n!, i=0..n) end do: G(k, x):= sum(S2[k, p]*((2*p)!/p!) *x^p/(1-4*x)^(p+1), p=0..k)/(((-1)^(k+1)*2*x)/(-1+4*x)^(k+1)): fx:=simplify(G(k, x)): nmax:=degree(fx); RHC[k-RHCnr+1]:= coeff(fx, x, k-RHCnr)/2^(k-RHCnr) end do: a:=n-> RHC[n]: seq(a(n), n=1..RHCend-RHCnr);

LHCnr:=5; LHCmax:=10: LHCend:=LHCnr+LHCmax: for k from LHCnr to LHCend do for n from 0 to k do S2[k, n]:=sum((-1)^(n+i)*binomial(n, i)*i^k/n!, i=0..n) end do: G(k, x):= sum(S2[k, p]*((2*p)!/p!)*x^p/(1-4*x)^(p+1), p=0..k)/ (((-1)^(k+1)*2*x)/(-1+4*x)^(k+1)): fx:=simplify(G(k, x)): nmax:=degree(fx); for n from 0 to nmax do d[n]:= coeff(fx, x, n)/2^n end do: LHC[n]:=d[LHCnr-1] end do: a:=n-> LHC[n]: seq(a(n), n=LHCnr..LHCend-1);

Originator sequences A142963, A156919.

Related sequences A156921, A156925, A156927, A156933.

Left hand column sequences A050488(n+1), A142965, A142966, A142968.

Right hand column sequences A000340, A156922, A156923, A156924.

Row sums (n) = A014307 (n+1).

Sequence in context: A157523 A141691 A157147 this_sequence A074060 A157637 A157181

Adjacent sequences: A156917 A156918 A156919 this_sequence A156921 A156922 A156923

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 2009