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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.

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A261358 Pentatope of coefficients in expansion of (1 + x + y + 2*z)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 4, 1, 4, 4, 1, 3, 3, 6, 3, 6, 12, 3, 12, 12, 1, 3, 6, 3, 12, 12, 1, 6, 12, 8, 1, 4, 4, 8, 6, 12, 24, 6, 24, 24, 4, 12, 24, 12, 48, 48, 4, 24, 48, 32, 1, 4, 8, 6, 24, 24, 4, 24, 48, 32, 1, 8, 24, 32, 16, 1, 5, 5, 10, 10, 20, 40, 10, 40, 40, 10, 30, 60, 30, 120, 120, 30, 120, 240, 160, 5, 40, 120, 160, 80, 1, 5, 10, 10, 40, 40, 10, 60, 120, 80, 5, 40, 120, 160, 80, 1, 10, 40, 80, 80, 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),(1,1,1,0) and two kinds of steps (1,1,1,1).
The sum of the numbers in each cell of the pentatope is 5^n (A000351).
The sum of the antidiagonals of each triangle in each slice gives A261357. - Dimitri Boscainos, Aug 21 2015

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 625. It is organized as follows:
.           1
.
.           4
.         4   8
.
.           6
.         12  24
.        6  24  24
.
.           4
.        12  24
.      12  48  48
.     4  24  48  32
.
.          1
.        4   8
.      6   24  24
.    4  24   48   32
.  1   8   24   32   16

Crossrefs

Cf. A000351, A189225, A261357, A261359, 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) +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

Formula

T(i+1,j,k,l) = 2*T(i,j-1,k-1,l-1) + T(i,j-1,k-1,l) + T(i,j-1,k,l) + T(i,j,k,l); a(i,j,k,-1)=0,...; a(0,0,0,0)=1.
T(n,i,j,k) = 2^k*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
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