A189225 -id:A189225 - cp's OEIS frontend

A261356 Pyramid of coefficients in expansion of (1+x+2*y)^n.

1, 1, 1, 2, 1, 2, 4, 1, 4, 4, 1, 3, 6, 3, 12, 12, 1, 6, 12, 8, 1, 4, 8, 6, 24, 24, 4, 24, 48, 32, 1, 8, 24, 32, 16, 1, 5, 10, 10, 40, 40, 10, 60, 120, 80, 5, 40, 120, 160, 80, 1, 10, 40, 80, 80, 32, 1, 6, 12, 15, 60, 60, 20, 120, 240, 160, 15, 120, 360, 480

Offset: 0

Dimitri Boscainos, Aug 16 2015

T(n,j,k) is the number of lattice paths from (0,0,0) to (n,j,k) with steps (1,0,0), (1,1,0) and two kinds of steps (1,1,1).
The sum of the terms in each slice of the pyramid is 4^n (A000302).
The terms of the j-th row of the n-th slice of this pyramid are the sum of the terms in each row of the j-th triangle of the n-th slice of A189225. - Dimitri Boscainos, Aug 21 2015

			Here is the fourth (n=3) slice of the pyramid:
.....1......
...3   6....
..3  12  12.
.1  6  12  8
As an irregular triangle, rows begin:
1;
1, 1, 2;
1, 2, 4, 1, 4, 4;
1, 3, 6, 3, 12, 12, 1, 6, 12, 8;
...

Alois P. Heinz, Rows n = 0..38, flattened

Cf. A046816, A261357.

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) +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);
# Alois P. Heinz, Aug 20 2015

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] + 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, Sep 16 2023, after Alois P. Heinz *)

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

T(n,j,k) = 2^k*binomial(n,j)*binomial(j,k). - Dimitri Boscainos, Aug 21 2015

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

Dimitri Boscainos, Aug 16 2015

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

			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

Cf. A000351, A189225, A261357, A261359, A261360.

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

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

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

Dimitri Boscainos, Aug 16 2015

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).

			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

Cf. A000400, A189225, A261358, A261360.

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

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

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 + 2x + 2y + 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

Dimitri Boscainos, Aug 16 2015

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).

			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

Cf. A000420, A189225, A261358, A261359.

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

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

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

A191358 Sequencing of all multinomial coefficients arranged in an s X r array of Pascal simplices P(s,r) and sequenced along the array's antidiagonals. Each P(s,r) is, in turn, a sequence of terms representing the coefficients of a_1,...,a_s in the expansion of (Sum_{i=1..s} a_i)^r with r starting at zero.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 4, 6, 4, 1, 1, 3, 3, 3, 6, 3, 1, 3, 3, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 10, 10, 5, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 3, 3, 3, 3, 6, 6, 3, 6, 3, 1, 3, 3, 3, 6, 3, 1, 3, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 15, 20, 15, 6, 1, 1, 5, 5, 10, 20, 10, 10, 30, 30, 10, 5, 20, 30, 20, 5, 1, 5, 10, 10, 5, 1, 1, 4, 4, 4, 6, 12, 12, 6, 12, 6, 4, 12, 12, 12, 24, 12, 4, 12, 12, 4, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 3, 3, 3, 3, 3, 6, 6, 6, 3, 6, 6, 3, 6, 3, 1, 3, 3, 3, 3, 6, 6, 3, 6, 3, 1, 3, 3, 3, 6, 3, 1, 3, 3, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1, 1, 5, 5, 5, 10, 20, 20, 10, 20, 10, 10, 30, 30, 30, 60, 30, 10, 30, 30, 10, 5, 20, 20, 30, 60, 30, 20, 60, 60, 20, 5, 20, 30, 20, 5, 1, 5, 5, 10, 20, 10, 10, 30, 30, 10, 5, 20, 30, 20, 5, 1, 5, 10, 10, 5, 1

Offset: 1

Frank M Jackson, May 31 2011

The Pascal simplices P(s,r) are sequenced along the s*r array's antidiagonals as P(1,0), P(1,1), P(2,0), P(1,2), P(2,1), P(3,0), P(1,3), P(2,2), P(3,1), P(4,0), etc. P(2,3) is the sequence 1,3,3,1. P(2,r) = Pascal's triangle = A007318. P(3,r) = Pascal's tetrahedron = A046816. P(4,r) = Pascal's 4D simplex = A189225. Each P(s,r) has binomial(s-1+r, s-1) terms. The sum of its terms is s^r. The Pascal simplex P(s,r) starts at a(n) where n = 2^(s+r-1) + Sum_{p=0..s-2} binomial(s+r-1,p).

			The Pascal simplex P(4,5) for the coefficients of (a_1 + a_2 + a_3 + a_4)^5 is the sequence:
.......1
.......5
......5,5
.......10
.....20,20
....10,20,10
.......10
.....30,30
....30,60,30
..10,30,30,10
.......5
.....20,20
....30,60,30
..20,60,60,20
..5 ,20,30,20,5
.......1
......5,5
....10,20,10
..10,30,30,10
.5, 20,30,20,5
1,5, 10,10, 5,1
The sequence starts at a(293), it has 56 terms and the sum of its terms is 1024. It is also the 40th Pascal simplex in the sequence counting along the antidiagonals of the s*r array of Pascal simpices P(s,r).
Within the Pascal simplex P(4,5) the term S(5,3,2,1) = binomial(5,3)*binomial(3,2)*binomial(2,1) = 60.

Wikipedia, Pascal's simplex.

p[s_, r_] := (f[t_] := Binomial[k[t - 1], k[t]] f[t - 1]; f[1] = 1;
  dim = s; k[1] = r; list = {}; vstring[0] = "{k[``],0,k[``]},";
  Do[vstring[i] = ToString[StringForm[vstring[0], i + 1, i]], {i, 1, dim - 1}];
  dostring = "Do[AppendTo[list,f[dim]],]";
  Do[dostring =
    StringInsert[dostring, vstring[j], StringLength[dostring]], {j, dim - 1}];
  dostring = StringDrop[dostring, {StringLength[dostring] - 1}];
  ToExpression[StringReverse@StringReplace[StringReverse@dostring, ","->"", 1]];
  Flatten[List[list]])
g[m_] := (For[h = 1; c = 1, c > 0, h++, c = m - h (h + 1)/2;
   a = m - h (h - 1)/2]; b = h - 1 - a; p[a, b])
Flatten[Table[g[e], {e, 1, 40}]]

The Pascal simplex P(s,r) starts at a(n) where n = 2^(s+r-1) + Sum_{p=0..s-2} binomial(s+r-1,p). The individual terms within the Pascal simplex, S(r,t_1,t_2,...,t_(s-1)) are given by S(r,t_1,t_2,...,t_(s-1)) = binomial(r,t_1)*binomial(t_1,t_2)*...*binomial(t_(s-2),t_(s-1)).

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A261356 Pyramid of coefficients in expansion of (1+x+2*y)^n.

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

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

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

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

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