A193789 -id:A193789 - cp's OEIS frontend

A193788 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=1+x^n.

1, 1, 1, 2, 1, 3, 4, 4, 1, 9, 8, 12, 6, 1, 27, 16, 32, 24, 8, 1, 81, 32, 80, 80, 40, 10, 1, 243, 64, 192, 240, 160, 60, 12, 1, 729, 128, 448, 672, 560, 280, 84, 14, 1, 2187, 256, 1024, 1792, 1792, 1120, 448, 112, 16, 1, 6561, 512, 2304, 4608, 5376, 4032, 2016

Offset: 0

Clark Kimberling, Aug 05 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

			First six rows:
1
1....1
2....1....3
4....4....1....9
8....12...6....1...27
16...32...24...8...1...81
 (viz., A038207 with row sums at end of rows)

Cf. A193722, A038207, A193789.

z = 10; a = 1; b = 2;
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}]]  (* A193788 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193789 *)

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

A193788 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=1+x^n.

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