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A156920 Triangle of the normalized A142963 and A156919 sequences. 18
1, 1, 1, 1, 5, 1, 1, 15, 18, 1, 1, 37, 129, 58, 1, 1, 83, 646, 877, 179, 1, 1, 177, 2685, 8030, 5280, 543, 1, 1, 367, 10002, 56285, 82610, 29658, 1636, 1, 1, 749, 34777, 335162, 919615, 756218, 159742, 4916, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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 A142963.

LINKS

Table of n, a(n) for n=0..44.

Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012

FORMULA

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.

From Peter Bala, Jul 22 2012: (Start)

T(n,k) = 1/(2^(n-k))*A156919(n,k).

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.

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).

(End)

EXAMPLE

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]

MAPLE

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);

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);

MATHEMATICA

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];

Table[T[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Nov 14 2017 *)

CROSSREFS

Originator sequences A142963, A156919.

Related sequences A156921, A156925, A156927, A156933.

Left hand column sequences A050488, A142965, A142966, A142968.

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

Row sums A014307(n+1).

Sequence in context: A157147 A232103 A292357 * A174044 A174159 A074060

Adjacent sequences:  A156917 A156918 A156919 * A156921 A156922 A156923

KEYWORD

easy,nonn,tabl

AUTHOR

Johannes W. Meijer, Feb 20 2009

EXTENSIONS

Minor edits by Johannes W. Meijer, Sep 28 2011

STATUS

approved

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Last modified October 20 17:53 EDT 2018. Contains 316397 sequences. (Running on oeis4.)