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.

A129186 Right shift operator generating 1's in shifted spaces.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Gary W. Adamson, Apr 01 2007

Keywords

Comments

Let A129186 = M, then M*V, V a vector; shifts V to the right, appending 1's to the shifted spaces. Example: M*V, V = [1,2,3,...] = [1,1,2,3,...].
Triangle T(n,k), read by rows, given by (1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 08 2011

Examples

			First few rows of the triangle are:
1;
1, 0;
0, 1, 0;
0, 0, 1, 0;
0, 0, 0, 1, 0;
...

Links

Crossrefs

Generalized Eulerian triangles: this sequence (m=0), A173018 (m=1), A292604 (m=2).
Cf. A000012 (row sums), A071919, A129184, A129185.

Programs

  • Maple
    gf := 1 + z/(1 - x*z): ser := series(gf, z, 16): c := k -> coeff(ser, z, k):
seq(seq(coeff(c(n), x, k), k=0..n), n=0..14); # Peter Luschny, Jul 07 2019
  • Mathematica
    Join[{1},Flatten[Table[PadLeft[{1,0},n,0],{n,2,20}]]] (* Harvey P. Dale, Aug 26 2019 *)

Formula

Infinite lower triangular matrix with (1,0,0,...) in the main diagonal and (1,1,1...) in the subdiagonal.
G.f.: (1-(y-1)*x)/(1-y*x). - Philippe Deléham, Dec 08 2011

A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

Original entry on oeis.org

1, 1, 0, 19, 1, 0, 1513, 166, 1, 0, 315523, 52715, 1361, 1, 0, 136085041, 30543236, 1528806, 10916, 1, 0, 105261234643, 29664031413, 2257312622, 42421946, 87375, 1, 0, 132705221399353, 45011574747714, 4637635381695, 153778143100, 1156669095, 699042, 1, 0
Offset: 0

Views

Author

Peter Luschny, Sep 20 2017

Keywords

Comments

See the comments in A292604.

Examples

			Triangle starts:
[n\k][       0         1         2       3  4  5]
--------------------------------------------------
[0][         1]
[1][         1,        0]
[2][        19,        1,        0]
[3][      1513,      166,        1,     0]
[4][    315523,    52715,     1361,     1,  0]
[5][ 136085041, 30543236,  1528806, 10916,  1, 0]

Crossrefs

F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} is this triangle, F_{4} = A292606.
First column: A002115. Row sums: A014606. Alternating row sums: A292609.
Cf. A181985, A278073.

Programs

  • Maple
    Coeffs := f -> PolynomialTools:-CoefficientList(expand(f),x):
A292605_row := proc(n) if n = 0 then return [1] fi;
add(A278073(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292605_row(n) od;
  • Sage
    # uses[A278073_row from A278073]
def A292605_row(n):
    if n == 0: return [1]
    L = A278073_row(n)
    S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..5): print(A292605_row(n))

Formula

F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) for n>0 and F_{3; 0}(x) = 1.

A292606 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).

Original entry on oeis.org

1, 1, 0, 69, 1, 0, 33661, 988, 1, 0, 60376809, 2669683, 16507, 1, 0, 288294050521, 17033188586, 212734266, 261626, 1, 0, 3019098162602349, 223257353561605, 4297382231090, 17634518610, 4196345, 1, 0
Offset: 0

Views

Author

Peter Luschny, Sep 26 2017

Keywords

Comments

See the comments in A292604.

Examples

			Triangle starts:
[n\k][          0            1          2       3   4   5]
--------------------------------------------------
[0] [           1]
[1] [           1,           0]
[2] [          69,           1,         0]
[3] [       33661,         988,         1,      0]
[4] [    60376809,     2669683,     16507,      1,  0]
[5] [288294050521, 17033188586, 212734266, 261626,  1,  0]

Crossrefs

F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} = A292605, F_{4} is this triangle.
First column: A211212. Row sums: A014608. Alternating row sums: A292607.
Cf. A181985.

Programs

  • Maple
    Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292606_row := proc(n) if n = 0 then return [1] fi;
add(A278074(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292606_row(n) od;
  • Sage
    # uses[A278074_row from A278074]
def A292606_row(n):
    if n == 0: return [1]
    L = A278074_row(n)
    S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..5): print(A292606_row(n))

Formula

F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) for n>0 and F_{4; 0}(x) = 1.
Showing 1-3 of 3 results.

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