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.

A261357 Pyramid of coefficients in expansion of (1 + 2*x + 2*y)^n.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 4, 4, 8, 4, 1, 6, 6, 12, 24, 12, 8, 24, 24, 8, 1, 8, 8, 24, 48, 24, 32, 96, 96, 32, 16, 64, 96, 64, 16, 1, 10, 10, 40, 80, 40, 80, 240, 240, 80, 80, 320, 480, 320, 80, 32, 160, 320, 320, 160, 32
Offset: 0

Views

Author

Dimitri Boscainos, Aug 16 2015

Keywords

Comments

T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), two kinds of steps (1,1,0) and two kinds of steps (1,1,1).
The sum of the numbers in each slice of the pyramid is 5^n.
The terms of the j-th row of the n-th slice of this pyramid are the sum of the terms in each antidiagonal of the j-th triangle of the n-th slice of A261358. - Dimitri Boscainos, Aug 21 2015

Examples

			Here is the fourth (n=3) slice of the pyramid:
        1
      6   6
   12  24  12
  8  24  24   8

Links

Crossrefs

Cf. A000079, A046816, A059304, A261356, A261358.

Programs

  • Maple
    p:= proc(i, j, k) option remember;
      if k<0 or i<0 or i>k or j<0 or j>i then 0
    elif {i, j, k}={0} then 1
    else p(i, j, k-1) +2*p(i-1, j, k-1) +2*p(i-1, j-1, k-1)
      fi
    end:
seq(seq(seq(p(i, j, k), j=0..i), i=0..k), k=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356
  • Mathematica
    p[i_, j_, k_] := p[i, j, k] = If[k < 0 || i < 0 || i > k || j < 0 || j > i, 0, If[Union@{i, j, k} == {0}, 1, p[i, j, k - 1] + 2*p[i - 1, j, k - 1] + 2*p[i - 1, j - 1, k - 1]]];
Table[Table[Table[p[i, j, k], {j, 0, i}], {i, 0, k}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Mar 17 2025, after Alois P. Heinz *)
  • PARI
    tabf(nn) = {for (n=0, nn, for (j=0, n, for (k=0, j, print1(2^j*binomial(n,j)*binomial(j,k), ", ")); print();); print(););} \\ Michel Marcus, Oct 07 2015

Formula

T(i+1,j,k) = 2*T(i,j-1,k-1)+ 2*T(i,j-1,k) + T(i,j,k); T(i,j,-1) = 0, ...; T(0,0,0) = 1.
T(n,j,k) = 2^j*binomial(n,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

A261359 Pentatope of coefficients in expansion of (1 + x + 2*y + 2*z)^n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 4, 4, 1, 4, 4, 4, 8, 4, 1, 3, 6, 6, 3, 12, 12, 12, 24, 12, 1, 6, 6, 12, 24, 12, 8, 24, 24, 8, 1, 4, 8, 8, 6, 24, 24, 24, 48, 24, 4, 24, 24, 48, 96, 48, 32, 96, 96, 32, 1, 8, 8, 24, 48, 24, 32, 96, 96, 32, 16, 64, 96, 64, 16, 1, 5, 10, 10, 10, 40, 40, 40, 80, 40, 10, 60, 60, 120, 240, 120, 80, 240, 240, 80, 5, 40, 40, 120, 240, 120, 160, 480, 480, 160, 80, 320, 480, 320, 80, 1, 10, 10, 40, 80, 40, 80, 240, 240, 80, 80, 320, 480, 320, 80, 32, 160, 320, 320, 160, 32
Offset: 0

Views

Author

Dimitri Boscainos, Aug 16 2015

Keywords

Comments

T(n,i,j,k) is the number of lattice paths from (0,0,0,0) to (n,i,j,k) with steps (1,0,0,0), (1,1,0,0) and two kinds of steps (1,1,1,0) and (1,1,1,1).
The sum of the numbers in each cell of the pentatope is 6^n (A000400).

Examples

			The 5th slice (n=4) of this 4D simplex starts at a(35). It comprises a 3D tetrahedron of 35 terms whose sum is 1296. It is organized as follows:
.
.           1
.
.           4
.         8   8
.
.           6
.        24  24
.      24  48  24
.
.           4
.        24  24
.      48  96  48
.    32  96  96  32
.
.           1
.         8   8
.      24  48  24
.    32  96  96  32
.  16  64  96  64  16

Crossrefs

Cf. A000400, A189225, A261358, A261360.

Programs

  • Maple
    p:= proc(i, j, k, l) option remember;
      if l<0 or j<0 or i<0 or i>l or j>i or k<0 or k>j then 0
    elif {i, j, k, l}={0} then 1
    else p(i, j, k, l-1) +p(i-1, j, k, l-1) +2*p(i-1, j-1, k, l-1)+2*p(i-1, j-1, k-1, l-1)
      fi
    end:
seq(seq(seq(seq(p(i, j, k, l), k=0..j), j=0..i), i=0..l), l=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356
  • PARI
    lista(nn) = {for (n=0, nn, for (i=0, n, for (j=0, i, for (k=0, j, print1(2^j*binomial(n,i)*binomial(i,j)*binomial(j,k), ", ")););););} \\ Michel Marcus, Oct 07 2015

Formula

T(i+1,j,k,l) = 2*T(i,j-1,k-1,l-1) + 2*T(i,j-1,k-1,l) + T(i,j-1,k,l) + T(i,j,k,l); T(i,j,k,-1)=0,...; T(0,0,0,0)=1.
T(n,i,j,k) = 2^j*binomial(n,i)*binomial(i,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

A261360 Pentatope of coefficients in expansion of (1 + 2*x + 2*y + 2*z)^n.

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 4, 4, 4, 4, 8, 8, 4, 8, 4, 1, 6, 6, 6, 12, 24, 24, 12, 24, 12, 8, 24, 24, 24, 48, 24, 8, 24, 24, 8, 1, 8, 8, 8, 24, 48, 48, 24, 48, 24, 32, 96, 96, 96, 192, 96, 32, 96, 96, 32, 16, 64, 64, 96, 192, 96, 64, 192, 192, 64, 16, 64, 96, 64, 96, 1, 10, 10, 10, 40, 80, 80, 40, 80, 40, 80, 240, 240, 240, 480, 240, 80, 240, 240, 80, 80, 320, 320, 480, 960, 480, 320, 960, 960, 320, 80, 320, 480, 320, 80, 32, 160, 160, 320, 640, 320, 320, 960, 960, 320, 160, 640, 960, 640, 160, 32, 160, 320, 320, 160, 32
Offset: 0

Views

Author

Dimitri Boscainos, Aug 16 2015

Keywords

Comments

T(n,i,j,k) is the number of lattice paths from (0,0,0,0) to (n,i,j,k) with steps (1,0,0,0) and two kinds of steps (1,1,0,0), (1,1,1,0) and (1,1,1,1).
The sum of the numbers in each cell of the pentatope is 7^n (A000420).

Examples

			The 5th slice (n=4) of this 4D simplex starts at a(35). It comprises a 3D tetrahedron of 35 terms whose sum is 2401. It is organized as follows:
.
.               1
.
.               8
.            8     8
.
.              24
.           48    48
.        24    48    24
.
.              32
.           96    96
.        96   192    96
.     32    96    96    32
.
.              16
.           64    64
.        96   192    96
.     64   192   192    64
.  16    64    96    64    16

Crossrefs

Cf. A000420, A189225, A261358, A261359.

Programs

  • Maple
    p:= proc(i, j, k, l) option remember;
      if l<0 or j<0 or i<0 or i>l or j>i or k<0 or k>j then 0
    elif {i, j, k, l}={0} then 1
    else p(i, j, k, l-1) +2*p(i-1, j, k, l-1) +2*p(i-1, j-1, k, l-1)+2*p(i-1, j-1, k-1, l-1)
      fi
    end:
seq(seq(seq(seq(p(i, j, k, l), k=0..j), j=0..i), i=0..l), l=0..5);
# Adapted from Alois P. Heinz's Maple program for A261356
  • PARI
    lista(nn) = {for (n=0, nn, for (i=0, n, for (j=0, i, for (k=0, j, print1(2^i*binomial(n,i)*binomial(i,j)*binomial(j,k), ", ")););););} \\ Michel Marcus, Oct 07 2015

Formula

T(i+1,j,k,l) = 2*T(i,j-1,k-1,l-1) + 2*T(i,j-1,k-1,l) + 2*T(i,j-1,k,l) + T(i,j,k,l); T(i,j,k,-1)=0, ...; T(0,0,0,0)=1.
T(n,i,j,k) = 2^i*binomial(n,i)*binomial(i,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.