cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A193824 Mirror of the triangle A193823.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 9, 9, 5, 1, 27, 27, 19, 7, 1, 81, 81, 65, 33, 9, 1, 243, 243, 211, 131, 51, 11, 1, 729, 729, 665, 473, 233, 73, 13, 1, 2187, 2187, 2059, 1611, 939, 379, 99, 15, 1, 6561, 6561, 6305, 5281, 3489, 1697, 577, 129, 17, 1, 19683, 19683, 19171
Offset: 0

Views

Author

Clark Kimberling, Aug 06 2011

Keywords

Comments

A193824 is obtained by reversing the rows of the triangle A193823.

Examples

			First six rows:
1
1....1
3....3....1
9....9....5.....1
27...27...19....7...1
81...81...65....33...9...1

Crossrefs

Cf. A193823.

Programs

  • Mathematica
    z = 10; a = 2; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; 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}]]   (* A193823 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193824 *)

Formula

Write w(n,k) for the triangle at A193823. The triangle at A193824 is then given by w(n,n-k).

A193860 Triangular array: the fission of ((2x+1)^n) by (q(n,x)), where q(n,x)=x^n+x^(n-1)+...+x+1.

Original entry on oeis.org

1, 1, 5, 1, 7, 19, 1, 9, 33, 65, 1, 11, 51, 131, 211, 1, 13, 73, 233, 473, 665, 1, 15, 99, 379, 939, 1611, 2059, 1, 17, 129, 577, 1697, 3489, 5281, 6305, 1, 19, 163, 835, 2851, 6883, 12259, 16867, 19171, 1, 21, 201, 1161, 4521, 12585, 26025, 41385, 52905
Offset: 0

Views

Author

Clark Kimberling, Aug 07 2011

Keywords

Comments

See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.

Examples

			First six rows:
1
1...5
1...7....19
1...9....33...65
1...11...51...131...211
1...13...73...233...473...665

Crossrefs

Cf. A193842, A193861. A193823.

Programs

  • Mathematica
    z = 10;
p[n_, x_] := (2 x + 1)^n;
q[0, x_] := 1; q[n_, x_] := x*q[n - 1, x] + 1;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193860 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193861  *)

Formula

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} binomial(n+1,k-i)*2^(k-i) for 0 <= k <= n.
O.g.f.: 1/( (1 - 3*x*t)*(1 - (2*x + 1)*t) ) = 1 + (1 + 5*x)*t + (1 + 7*x + 19*x^2)*t^2 + ....
The n-th row polynomial R(n,x) = 1/(1 - x)*( (2*x + 1)^(n+1) - (3*x)^(n+1) ). Cf. A193823. (End)

A193820 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 7, 8, 8, 1, 5, 11, 15, 16, 16, 1, 6, 16, 26, 31, 32, 32, 1, 7, 22, 42, 57, 63, 64, 64, 1, 8, 29, 64, 99, 120, 127, 128, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256, 256, 1, 10, 46, 130, 256, 382, 466, 502, 511, 512, 512, 1, 11, 56
Offset: 0

Views

Author

Clark Kimberling, Aug 06 2011

Keywords

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
Variant of A054143 and A008949. - R. J. Mathar, Mar 03 2013

Examples

			First six rows:
  1
  1....1
  1....2....2
  1....3....4....4
  1....4....7....8....8
  1....5....11...15...16...16

Crossrefs

Cf. A193722, A128175, A193823, A045623 (row sums), A009766.
Cf. A054143, A008949.

Programs

  • Maple
    A193820 := (n,k) -> `if`(k=0 or n=0,1, A193820(n-1,k-1)+A193820(n-1,k));
seq(print(seq(A193820(n,k),k=0..n+1)),n=0..10); # Peter Luschny, Jan 22 2012
  • Mathematica
    z = 10; a = 1; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; 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}]]   (* A193820 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A128175 *)

Formula

From Peter Bala, Jul 16 2013: (Start)
T(n,k) = sum {i = 0..k} binomial(n-1,k-i) for 0 <= k <= n.
O.g.f.: (1 - x*t)^2/( (1 - 2*x*t)*(1 - (1 + x)*t) ) = 1 + (1 + x)*t + (1 + 2*x + 2*x^2)*t^2 + ....
The n-th row polynomial R(n,x) for n >= 1 is given by R(n,x) = 1/(1 - x)*( (x + 1)^(n-1) - 2^(n-1)*x^(n+1) ). Cf. A193823. (End)
Showing 1-3 of 3 results.

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