A033543 -id:A033543 - cp's OEIS frontend

A052179 Triangle of numbers arising in enumeration of walks on cubic lattice.

1, 4, 1, 17, 8, 1, 76, 50, 12, 1, 354, 288, 99, 16, 1, 1704, 1605, 700, 164, 20, 1, 8421, 8824, 4569, 1376, 245, 24, 1, 42508, 48286, 28476, 10318, 2380, 342, 28, 1, 218318, 264128, 172508, 72128, 20180, 3776, 455, 32, 1, 1137400, 1447338

Offset: 0

N. J. A. Sloane, Jan 26 2000

Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 4*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007
Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and four types of steps H=(1,0); example: T(3,1)=50 because we have UDU, UUD, 16 HHU paths, 16 HUH paths and 16 UHH paths. - Philippe Deléham, Sep 25 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007
Riordan array ((1-4x-sqrt(1-8x+12x^2))/(2x^2), (1-4x-sqrt(1-8x+12x^2))/(2x)). Inverse of A159764. - Paul Barry, Apr 21 2009
6^n = (n-th row terms) dot (first n+1 terms in (1,2,3,...)). Example: 6^3 =  216 = (76, 50, 12, 1) dot (1, 2, 3, 4) = (76 + 100 + 36 + 4) = 216. - Gary W. Adamson, Jun 15 2011
A subset of the "family of triangles" (Deléham comment of Sep 25 2007) is the succession of binomial transforms beginning with triangle A053121, (0,0); giving -> A064189, (1,1); -> A039598, (2,2); -> A091965, (3,3); -> A052179, (4,4); -> A125906, (5,5) ->, etc.; generally the binomial transform of the triangle generated from (n,n) = that generated from ((n+1),(n+1)). - Gary W. Adamson, Aug 03 2011

			Triangle begins:
    1;
    4,   1;
   17,   8,   1;
   76,  50,  12,   1;
  354, 288,  99,  16,   1;
  ...
Production matrix begins:
  4, 1;
  1, 4, 1;
  0, 1, 4, 1;
  0, 0, 1, 4, 1;
  0, 0, 0, 1, 4, 1;
  0, 0, 0, 0, 1, 4, 1;
  0, 0, 0, 0, 0, 1, 4, 1;
- _Philippe Deléham_, Nov 04 2011

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6.

Cf. A039598, A053121, A064189, A091965.

T:= proc(n, k) option remember; `if`(min(n, k)<0, 0,
     `if`(max(n, k)=0, 1, T(n-1, k-1)+4*T(n-1, k)+T(n-1, k+1)))
    end:
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 28 2021

t[0, 0] = 1; t[n_, k_] /; k < 0 || k > n = 0; t[n_, 0] := t[n, 0] = 4*t[n-1, 0] + t[n-1, 1]; t[n_, k_] := t[n, k] = t[n-1, k-1] + 4*t[n-1, k] + t[n-1, k+1]; Flatten[ Table[t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Oct 10 2011, after Philippe Deleham *)

Sum_{k>=0} T(m, k)*T(n, k) = T(m+n, 0) = A005572(m+n). - Philippe Deléham, Sep 15 2005
n-th row = M^n * V, where M = the infinite tridiagonal matrix with all 1's in the super and subdiagonals and (4,4,4,...) in the main diagonal. E.g., Row 3 = (76, 50, 12, 1) since M^3 * V = [76, 50, 12, 1, 0, 0, 0, ...]. - Gary W. Adamson, Nov 04 2006
Sum_{k=0..n} T(n,k) = A005573(n). - Philippe Deléham, Feb 04 2007
Sum_{k=0..n} T(n,k)*(k+1) = 6^n. - Philippe Deléham, Mar 27 2007
Sum_{k=0..n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x = -2, -1, 0, 1 respectively. - Philippe Deléham, Nov 28 2009
As an infinite lower triangular matrix = the binomial transform of A091965 and 4th binomial transform of A053121. - Gary W. Adamson, Aug 03 2011
G.f.: 2/(1 - 4*x - 2*x*y + sqrt(1 - 8*x + 12*x^2)). - Daniel Checa, Aug 17 2022
G.f. for the m-th column: x^m*(A(x))^(m+1), where A(x) is the g.f. of the sequence counting the walks on the cubic lattice starting and finishing on the xy plane and never going below it (A005572). Explicitly, the g.f. is x^m*((1 - 4*x - sqrt(1 - 8*x + 12*x^2))/(2*x^2))^(m+1). - Daniel Checa, Aug 28 2022

A124575 Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (2,4,4,...) and super- and subdiagonals (1,1,1,...).

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 16, 30, 10, 1, 62, 146, 71, 14, 1, 270, 717, 444, 128, 18, 1, 1257, 3582, 2621, 974, 201, 22, 1, 6096, 18206, 15040, 6718, 1800, 290, 26, 1, 30398, 93960, 85084, 43712, 14208, 2986, 395, 30, 1, 154756, 491322, 478008, 274140, 103530

Offset: 0

Gary W. Adamson & Roger L. Bagula, Nov 05 2006

Column k=0 yields A033543 (2nd binomial transform of the sequence A000957(n+1)). Row sums yield A133158. [Corrected by Philippe Deléham, Oct 24 2007, Dec 05 2009]
Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 2*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Mar 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1)  +y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

			Row 2 is (5,6,1) because M[3]= [2,1,0;1,4,1;0,1,4] and M[3]^2=[5,6,1;6,18,8;1,8,17].
Triangle starts:
    1;
    2,   1;
    5,   6,   1;
   16,  30,  10,   1;
   62, 146,  71,  14,  1;
  270, 717, 444, 128, 18, 1;

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A124576, A124574, A052179, A064613, A133158.

with(linalg): m:=proc(i,j) if i=1 and j=1 then 2 elif i=j then 4 elif abs(i-j)=1 then 1 else 0 fi end: for n from 3 to 11 do A[n]:=matrix(n,n,m): B[n]:=multiply(seq(A[n],i=1..n-1)) od: 1; 2,1; for n from 3 to 11 do seq(B[n][1,j],j=1..n) od; # yields sequence in triangular form

M[n_] := SparseArray[{{1, 1} -> 2, Band[{2, 2}] -> 4, Band[{1, 2}] -> 1, Band[{2, 1}] -> 1}, {n, n}]; row[1] = {1}; row[n_] := MatrixPower[M[n], n-1] // First // Normal; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)

T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k-1) for k >= 2.
Sum_{k=0..n} T(n,k)*(3*k+1) = 6^n. - Philippe Deléham, Mar 27 2007
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A033543(m+n). - Philippe Deléham, Nov 22 2009

Edited by N. J. A. Sloane, Dec 04 2006

A171568 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A064613.

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 37, 29, 9, 1, 150, 134, 57, 12, 1, 654, 622, 318, 94, 15, 1, 3012, 2948, 1686, 616, 140, 18, 1, 14445, 14317, 8781, 3693, 1055, 195, 21, 1, 71398, 71142, 45625, 21132, 7075, 1662, 259, 24, 1, 361114, 360602, 238170, 118042, 44303, 12345, 2464, 332, 27, 1

Offset: 0

Philippe Deléham, Dec 11 2009

Equal to A171515*B = B*A104259, B = A007318.

			Triangle T(n,k) begins
[0]     1;
[1]     3,     1;
[2]    10,     6,     1;
[3]    37,    29,     9,     1;
[4]   150,   134,    57,    12,    1;
[5]   654,   622,   318,    94,   15,    1;
[6]  3012,  2948,  1686,   616,  140,   18,   1;
[7] 14445, 14317,  8781,  3693, 1055,  195,  21,  1;
[8] 71398, 71142, 45625, 21132, 7075, 1662, 259, 24, 1;
.
Production array begins
  3, 1
  1, 3, 1
  1, 1, 3, 1
  1, 1, 1, 3, 1
  1, 1, 1, 1, 3, 1
  1, 1, 1, 1, 1, 3, 1
- _Philippe Deléham_, Mar 05 2013

Sum_{k=0..n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x = -1, 0, 1, 2 respectively.

Cf. A007318, A052179, A171589, A171515, A104259.

T := proc(n,k) option remember;
if n < 0 or k < 0 then 0 elif n = k then 1 else
T(n-1, k-1) + 3*T(n-1,k) + add(T(n-1, k+1+i), i=0..n) fi end:
for n from 0 to 8 do seq(T(n,k), k = 0..n) od; # Peter Luschny, Oct 16 2022

T[n_, k_] := T[n, k] = If[n < 0 || k < 0, 0, If[n == k, 1, T[n-1, k-1] + 3*T[n-1, k] + Sum[T[n-1, k+1+i], {i, 0, n}]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 23 2024, after Peter Luschny *)

T(n, 0) - T(n, 1) = 2^n.

T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + Sum_{i=0..n} T(n-1, k+1+i). - Philippe Deléham, Feb 23 2012

Corrected and extended by Peter Luschny, Oct 16 2022

cp's OEIS Frontend

A171515 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A033543.

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A052179 Triangle of numbers arising in enumeration of walks on cubic lattice.

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A124575 Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (2,4,4,...) and super- and subdiagonals (1,1,1,...).

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A171568 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A064613.

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A171388 Expansion of the first column of triangle T_(2,x), T_(x,y) defined in A039599; T_(2,0)= A126075, T_(2,1)= A038622, T_(2,2)= A039598, T_(2,3)= A124733, T_(2,4)= A124575.

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A171616 Triangle T : T(n,k)= binomial(n,k)*A000957(n+1-k).

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