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A193819 Mirror of the triangle A193818. 2
1, 1, 2, 2, 6, 4, 3, 12, 16, 8, 4, 20, 40, 40, 16, 5, 30, 80, 120, 96, 32, 6, 42, 140, 280, 336, 224, 64, 7, 56, 224, 560, 896, 896, 512, 128, 8, 72, 336, 1008, 2016, 2688, 2304, 1152, 256, 9, 90, 480, 1680, 4032, 6720, 7680, 5760, 2560, 512, 10, 110, 660 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A193819 is obtained by reversing the rows of the triangle A193818.

LINKS

Table of n, a(n) for n=0..57.

FORMULA

Write w(n,k) for the triangle at A193818.  The triangle at A193819 is then given by w(n,n-k).

Triangle T(n,k), read by rows, given by (1,1,-1,1,0,0,0,0,0,0,0,...) DELTA (2,0,-2,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

T(n,k) = A153861(n,k)*2^k. - Philippe Deléham, Oct 09 2011

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-1), T(0,0)=T(1,0)=1, T(1,1)=T(2,0)=2, T(2,1)=6, T(2,2)=4, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Dec 15 2013

G.f.: (1-x+x^2+2*x^2*y)/((x-1)*(-1+x+2*x*y)). - R. J. Mathar, Aug 12 2015

EXAMPLE

First six rows:

  1;

  1,   2;

  2,   6,   4;

  3,  12,  16,   8;

  4,  20,  40,  40,  16;

  5,  30,  80, 120,  96,  32;

MATHEMATICA

z = 10; c = 2; d = 1;

p[0, x_] := 1

p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;

q[n_, x_] := (c*x + d)^n

t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

g[n_] := CoefficientList[w[n, x], {x}]

TableForm[Table[Reverse[g[n]], {n, -1, z}]]

Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193818 *)

TableForm[Table[g[n], {n, -1, z}]]

Flatten[Table[g[n], {n, -1, z}]]   (* A193819 *)

CROSSREFS

Cf. A084938, A193722, A193818.

Sequence in context: A064851 A305353 A134458 * A182786 A009279 A059943

Adjacent sequences:  A193816 A193817 A193818 * A193820 A193821 A193822

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 06 2011

STATUS

approved

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)