A193908 -id:A193908 - cp's OEIS frontend

A193909 Mirror of the triangle A193908.

1, 1, 2, 3, 6, 8, 6, 12, 20, 24, 11, 22, 40, 64, 80, 19, 38, 72, 128, 208, 256, 32, 64, 124, 232, 416, 672, 832, 53, 106, 208, 400, 752, 1344, 2176, 2688, 87, 174, 344, 672, 1296, 2432, 4352, 7040, 8704, 142, 284, 564, 1112, 2176, 4192, 7872, 14080, 22784

Offset: 0

Clark Kimberling, Aug 09 2011

A193909 is obtained by reversing the rows of the triangle A193908.

			First six rows:
1
1....2
3....6....8
6....12...20...24
11...22...40...64....80
19...38...72...128...208...256

Cf. A193908.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 2]*x^(n - k), {k, 0, n}];
q[n_, x_] := 2 x*q[n - 1, x] + 1 ; q[0, x_] := 1;
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}]]  (* A193908 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193909 *)

Write w(n,k) for the triangle at A193908. The triangle at A193909 is then given by w(n,n-k).

A193915 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2xq(n-1,x)+1 with q(0,x)=1.

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 16, 12, 8, 4, 48, 40, 24, 14, 7, 160, 128, 80, 44, 24, 12, 512, 416, 256, 144, 76, 40, 20, 1664, 1344, 832, 464, 248, 128, 66, 33, 5376, 4352, 2688, 1504, 800, 416, 212, 108, 54, 17408, 14080, 8704, 4864, 2592, 1344, 688, 348, 176, 88

Offset: 0

Clark Kimberling, Aug 09 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
First five rows of Q:
1
2....1
4....2...1
8....4...2...1
16...8...4...2...1

			First six rows:
1
2....1
4....4....2
16...12...8...4
48...40...24..14..7
160..128..80..44..24..12

Cf. A193722, A193916, A193908.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := 2 x*q[n - 1, x] + 1 ; q[0, x_] := 1;
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}]] (* A193915 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193916 *)

A193916 Mirror of the triangle A193915.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 4, 8, 12, 16, 7, 14, 24, 40, 48, 12, 24, 44, 80, 128, 160, 20, 40, 76, 144, 256, 416, 512, 33, 66, 128, 248, 464, 832, 1344, 1664, 54, 108, 212, 416, 800, 1504, 2688, 4352, 5376, 88, 176, 348, 688, 1344, 2592, 4864, 8704, 14080, 17408

Offset: 0

Clark Kimberling, Aug 09 2011

A193916 is obtained by reversing the rows of the triangle A193915.

			First six rows:
1
1....2
2....4....4
4....8....12...16
7....14...24...40...48
12...24...44...80...128...160

Cf. A193915, A193909.

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 2]*x^(n - k), {k, 0, n}];
q[n_, x_] := 2 x*q[n - 1, x] + 1 ; q[0, x_] := 1;
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}]]  (* A193908 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193909 *)

Write w(n,k) for the triangle at A193915. The triangle at A193916 is then given by w(n,n-k).

cp's OEIS Frontend

A193909 Mirror of the triangle A193908.

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A193915 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2xq(n-1,x)+1 with q(0,x)=1.

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A193916 Mirror of the triangle A193915.

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cp's OEIS Frontend

A193909 Mirror of the triangle A193908.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A193915 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2x*q(n-1,x)+1 with q(0,x)=1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A193916 Mirror of the triangle A193915.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A193915 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=sum{F(k+1)x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers), and q(n,x)=2xq(n-1,x)+1 with q(0,x)=1.