A193790 -id:A193790 - cp's OEIS frontend

A193791 Mirror of the triangle A193790.

1, 1, 1, 3, 2, 1, 9, 4, 4, 1, 27, 8, 12, 6, 1, 81, 16, 32, 24, 8, 1, 243, 32, 80, 80, 40, 10, 1, 729, 64, 192, 240, 160, 60, 12, 1, 2187, 128, 448, 672, 560, 280, 84, 14, 1, 6561, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 19683, 512, 2304, 4608, 5376, 4032

Offset: 0

Clark Kimberling, Aug 05 2011

A193791 is obtained by reversing the rows of the triangle A193790.

			First six rows:
1
1....1
3....2....1
9....4....4....1
27..8....12....6...1
81...16...32....24...8....1

Cf. A193790.

z = 10; a = 2; b = 1;
p[n_, x_] := (a*x + b)^n
q[n_, x_] := 1 + x^n ; q[n_, 0] := q[n, x] /. x -> 0;
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}]]  (* A193790 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193791 *)

Write w(n,k) for the triangle at A193790. The triangle at A193791 is then given by w(n,n-k).

A182059 Triangle, read by rows, given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 4, 4, 0, 6, 12, 8, 0, 8, 24, 32, 16, 0, 10, 40, 80, 80, 32, 0, 12, 60, 160, 240, 192, 64, 0, 14, 84, 280, 560, 672, 448, 128, 0, 16, 112, 448, 1120, 1792, 1792, 1024, 256, 0, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512

Offset: 0

Philippe Deléham, Apr 09 2012

Row sums are 3^n - 1 + 0^n.

Triangle of coefficients in expansion of (1+2*x)^n - 1 + 0^n .

			Triangle begins :
1
0, 2
0, 4, 4
0, 6, 12, 8
0, 8, 24, 32, 16
0, 10, 40, 80, 80, 32
0, 12, 60, 160, 240, 192, 64
0, 14, 84, 280, 560, 672, 448, 128
0, 16, 112, 448, 1120, 1792, 1792, 1024, 256
0, 18, 144, 672, 2016, 4032, 5376, 4608, 2304, 512
0, 20, 180, 960, 3360, 8064, 13440, 15360, 11520, 5120, 1024

Cf. A000079, A007318, A013609, A193789, A193790

G.f.: (1-2*x+x^2+2*y*x^2)/(1-2*x-2*y*x+x^2+2*y*x^2).
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) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,1) = T(2,2) = 4 and T(n,k) = 0 if k<0 or if k>n.
T(n,k) = A206735(n,k)*2^k.
T(n,k) = A013609(n,k) - A073424(n,k) .

cp's OEIS Frontend

A193791 Mirror of the triangle A193790.

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