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%I #19 Apr 15 2021 10:13:53
%S 1,1,1,1,5,1,1,15,18,1,1,37,129,58,1,1,83,646,877,179,1,1,177,2685,
%T 8030,5280,543,1,1,367,10002,56285,82610,29658,1636,1,1,749,34777,
%U 335162,919615,756218,159742,4916,1
%N Triangle of the normalized A142963 and A156919 sequences.
%C The originator sequences are A142963 and A156919.
%C 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.
%C The first Maple program generates the Flower Triangle sequence.
%C 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 A142963.
%H Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012
%F T(n,m) = (m+1)*T(n-1,m) + (2*n-2*m+1)*T(n-1,m-1) with T(n,m=0) = 1 and T(n,n) = 1, n>=0 and 0 <= m <= n.
%F From _Peter Bala_, Jul 22 2012: (Start)
%F T(n,k) = 1/(2^(n-k))*A156919(n,k).
%F E.g.f.: 1 + t*x + (t+t^2)*x^2/2! + (t+5*t^2+t^3)*x^3/3! + ... = sqrt(E(x,2*t)), where E(x,t) = (1-t)*exp(x*t)/(exp(x*t)-t*exp(x)) = 1 + t*x + (t+t^2)*x^2/2! + (t+4*t^2+t^3)*x^3/3! + ... is the e.g.f. for the Eulerian numbers A008292.
%F The row polynomials R(n,x) satisfy 1/sqrt(1-2*x)*(x*d/dx)^n(1/sqrt(1-2*x)) = R(n,x)/(1-2*x)^(n+1).
%F (End)
%e The first few rows of the triangle are:
%e [1]
%e [1, 1]
%e [1, 5 , 1 ]
%e [1, 15, 18, 1]
%e [1, 37, 129, 58, 1]
%e [1, 83, 646, 877, 179, 1]
%p A156920 := proc(n,m): if n=m then 1; elif m=0 then 1 ; elif m<0 or m>n then 0; else (m+1)*procname(n-1, m)+(2*n-2*m+1)*procname(n-1, m-1) ; end if; end proc: seq(seq(A156920(n, m), m=0..n), n=0..8);
%p 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);
%p 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);
%t T[_, 0] = 1; T[n_, n_] = 1; T[n_, m_] := T[n, m] = (m + 1)*T[n - 1, m] + (2*n - 2*m + 1)*T[n - 1, m - 1];
%t Table[T[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 14 2017 *)
%Y Originator sequences A142963, A156919.
%Y Related sequences A156921, A156925, A156927, A156933.
%Y Left hand column sequences A050488, A142965, A142966, A142968.
%Y Right hand column sequences A000340, A156922, A156923, A156924.
%Y Row sums A014307(n+1).
%K easy,nonn,tabl
%O 0,5
%A _Johannes W. Meijer_, Feb 20 2009
%E Minor edits by Johannes W. Meijer, Sep 28 2011
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