A038622 -id:A038622 - cp's OEIS frontend

A124733 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 (2,3,3,...) and super- and subdiagonals (1,1,1,...).

1, 2, 1, 5, 5, 1, 15, 21, 8, 1, 51, 86, 46, 11, 1, 188, 355, 235, 80, 14, 1, 731, 1488, 1140, 489, 123, 17, 1, 2950, 6335, 5397, 2730, 875, 175, 20, 1, 12235, 27352, 25256, 14462, 5530, 1420, 236, 23, 1, 51822, 119547, 117582, 74172, 32472, 10026, 2151, 306, 26, 1

Offset: 1

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

With a different offset: 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)+3*T(n-1,k)+T(n-1,k+1) for k>=1. - Philippe Deléham, Mar 27 2007
Equals A007318*A039599 (when written as lower triangular matrix). - Philippe Deléham, Jun 16 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
5^n = (n-th row terms) dot (first n+1 odd integers). Example: 5^4 = 625 = (51, 86, 46, 11, 1) dot (1, 3, 5, 7, 9) = (51 + 258 + 230 + 77 + 9) = 625. [Gary W. Adamson, Jun 13 2011]

			Row 3 is (5,5,1) because M[3]=[2,1,0;1,3,1;0,1,3] and M[3]^2=[5,5,1;5,11,6;1,6,10].
Triangle starts:
1;
2, 1;
5, 5, 1;
15, 21, 8, 1;
51, 86, 46, 11, 1;
188, 355, 235, 80, 14, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Shu-Chiuan Chang and Robert Shrock, Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips, Journal of Statistical Physics 137 (2009) 667.

Cf. A110877, A091965, A002212, A007317, A026375 (row sums).

with(linalg): m:=proc(i,j) if i=1 and j=1 then 2 elif i=j then 3 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
T := (n,k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k,-n+1,3/2) + GegenbauerC(n-k-1,-n+1,3/2)): seq(seq(T(n,k), k=1..n), n=1..10); # Peter Luschny, May 13 2016

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,  T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 2, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

Sum_{k=0..n} (-1)^(n-k)*T(n,k) = (-1)^n. - Philippe Deléham, Feb 27 2007
Sum_{k=0..n} T(n,k)*(2*k+1) = 5^n. - Philippe Deléham, Mar 27 2007
T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,3/2) + GegenbauerC(n-k-1,-n+1,3/2)). - Peter Luschny, May 13 2016
From Peter Bala, Sep 06 2022: (Start)
The following assume the row and column indexing start at 0.
Riordan array (f(x), x*g(x)), where f(x) = ( 1 - sqrt((1 - 5*x)/(1 - x)) )/(2*x) = 1 + 2*x + 5*x^2 + 15*x^3 + 51*x^4 + ... is the o.g.f. of A007317 and g(x) = ( 1 - 3*x - sqrt(1 - 6*x + 5*x^2) )/(2*x^2) = 1 + 3*x + 10*x^2 + 36*x^3 + 137*x^4 + .... See A002212.
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x)*(1 + 3*x + x^2)^n expanded about the point x = 0.
T(n,k) = a(n,k) - a(n,k+1), where a(n,k) = Sum_{j = 0..n} binomial(n,j)* binomial(j,n-k-j)*3^(2*j+k-n). (End)

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

A124576 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 (1,4,4,...) and super- and subdiagonals (1,1,1,...).

Original entry on oeis.org

1, 1, 1, 2, 5, 1, 7, 23, 9, 1, 30, 108, 60, 13, 1, 138, 522, 361, 113, 17, 1, 660, 2587, 2079, 830, 182, 21, 1, 3247, 13087, 11733, 5581, 1579, 267, 25, 1, 16334, 67328, 65600, 35636, 12164, 2672, 368, 29, 1, 83662, 351246, 365364, 220308, 86964, 23220, 4173

Offset: 1

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

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)=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 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

			Row 3 is (2,5,1) because M[3]=[1,1,0;1,4,1;0,1,4] and M[3]^2=[2,5,1;5,18,8;1,8,17].
Triangle starts:
1;
1, 1;
2, 5, 1;
7, 23, 9, 1;
30, 108, 60, 13, 1;
138, 522, 361, 113, 17, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A124575, A124574, A052179, A227081 (row sums).

with(linalg): m:=proc(i,j) if i=1 and j=1 then 1 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; 1,1; for n from 3 to 11 do seq(B[n][1,j],j=1..n) od; # yields sequence in triangular form
# alternative
A124576_row := proc(n)
    if n = 0 then
        return [1] ;
    else
        M := Matrix(n,n) ;
        M[1,1] := 1;
        for c from 2 to n do
            if c = 2 then
                M[1,c] := 1;
            else
                M[1,c] := 0;
            end if;
        end do:
        for r from 2 to n do
            for c from 1 to n do
                if r = c then
                    M[r,c] := 4;
                elif abs(r-c) = 1 then
                    M[r,c] := 1;
                else
                    M[r,c] := 0;
                end if;
            end do:
        end do:
        LinearAlgebra[MatrixPower](M,n-1) ;
        return [seq(%[1,r],r=1..n)] ;
    end if;
end proc:
for n from 0 to 10 do
    A124576_row(n) ;
    print(%) ;
end do: # R. J. Mathar, May 20 2025

M[n_] := SparseArray[{{1, 1} -> 1, 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 *)

Sum_{k=0..n} T(n,k)*(4*k+1) = 6^n. - Philippe Deléham, Mar 27 2007

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

A126970 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) = T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1.

Original entry on oeis.org

1, 0, 1, 1, 3, 1, 3, 11, 6, 1, 11, 42, 30, 9, 1, 42, 167, 141, 58, 12, 1, 167, 684, 648, 327, 95, 15, 1, 684, 2867, 2955, 1724, 627, 141, 18, 1, 2867, 12240, 13456, 8754, 3746, 1068, 196, 21, 1, 12240, 53043, 61362, 43464, 21060, 7146, 1677, 260, 24, 1

Offset: 0

Philippe Deléham, Mar 19 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

			Triangle begins:
    1;
    0,   1;
    1,   3,   1;
    3,  11,   6,   1;
   11,  42,  30,   9,  1;
   42, 167, 141,  58, 12,  1;
  167, 684, 648, 327, 95, 15, 1; ...
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  0, 1
  1, 3, 1
  0, 1, 3, 1
  0, 0, 1, 3, 1
  0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 0, 1, 3, 1 (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,  T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 0, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

Sum_{k=0..n} T(n,k) = A126952(n).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A117641(m+n).
Sum_{k=0..n} T(n,k)*(4*k+1) = 5^n. - Philippe Deléham, Mar 22 2007

A124574 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 (3,4,4,...) and super- and subdiagonals (1,1,1,...).

Original entry on oeis.org

1, 3, 1, 10, 7, 1, 37, 39, 11, 1, 150, 204, 84, 15, 1, 654, 1050, 555, 145, 19, 1, 3012, 5409, 3415, 1154, 222, 23, 1, 14445, 28063, 20223, 8253, 2065, 315, 27, 1, 71398, 146920, 117208, 55300, 16828, 3352, 424, 31, 1, 361114, 776286, 671052, 355236, 125964, 30660, 5079, 549, 35, 1

Offset: 1

Gary W. Adamson & Roger L. Bagula, Nov 04 2006

Column 1 yields A064613. Row sums yield A081671.
Triangle T(n,k), 0 <= k <= n, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*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). - Philippe Deléham, Feb 27 2007
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) = 3*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
6^n = ((n+1)-th row terms) dot (first n+1 odd integers). Example: 6^4 = 1296 = (150, 204, 84, 15, 1) dot (1, 3, 5, 7, 9) = (150 + 612 + 420 + 105 + 9)= 1296. - Gary W. Adamson, Jun 15 2011
From Peter Bala, Sep 06 2022: (Start)
The following assume the row and column indexing start at 0.
Riordan array (f(x), x*g(x)), where f(x) = (1 - sqrt((1 - 6*x)/(1 - 2*x)))/(2*x) is the o.g.f. of A064613 and g(x) =  (1 - 4*x - sqrt(1 - 8*x + 12*x^2))/(2*x^2) is the o.g.f. of A005572.
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x)*(1 + 4*x + x^2)^n expanded about the point x = 0.
T(n,k) = a(n,k) - a(n,k+1), where a(n,k) = Sum_{j = 0..n} binomial(n,j)* binomial(j,n-k-j)*4^(2*j+k-n). (End)

			Row 4 is (37,39,11,1) because M[4]= [3,1,0,0;1,4,1,0;0,1,4,1;0,0,1,4] and M[4]^3=[37,39,11,1; 39, 87, 51, 12; 11, 51, 88, 50; 1, 12, 50, 76].
Triangle starts:
    1;
    3,    1
   10,    7,   1;
   37,   39,  11,   1
  150,  204,  84,  15,  1;
  654, 1050, 555, 145, 19, 1;
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  3, 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
  0, 0, 0, 0, 0, 0, 1, 4, 1
  0, 0, 0, 0, 0, 0, 0, 1, 4, 1 (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000108, A081671 (row sums), A124575, A124576, A052179, A064613, A005572.

with(linalg): m:=proc(i,j) if i=1 and j=1 then 3 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; 3,1; for n from 3 to 11 do seq(B[n][1,j],j=1..n) od; # yields sequence in triangular form
T := (n,k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k,-n+1,2)+GegenbauerC(n-k-1,-n+1,2 )): seq(print(seq(T(n,k),k=1..n)), n=1..10); # Peter Luschny, May 13 2016

M[n_] := SparseArray[{{1, 1} -> 3, 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[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]]; Table[T[n, k, 3, 4], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Sum_{k=0..n} (-1)^(n-k)*T(n,k) = (-2)^n. - Philippe Deléham, Feb 27 2007
Sum_{k=0..n} T(n,k)*(2*k+1) = 6^n. - Philippe Deléham, Mar 27 2007
T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,2) + GegenbauerC(n-k-1,-n+1,2)). - Peter Luschny, May 13 2016

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

A126331 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 4T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 5T(n-1,k) + T(n-1,k+1) for k >= 1.

Original entry on oeis.org

1, 4, 1, 17, 9, 1, 77, 63, 14, 1, 371, 406, 134, 19, 1, 1890, 2535, 1095, 230, 24, 1, 10095, 15660, 8240, 2269, 351, 29, 1, 56040, 96635, 59129, 19936, 4053, 497, 34, 1, 320795, 598344, 412216, 162862, 40698, 6572, 668, 39, 1

Offset: 0

Philippe Deléham, Mar 10 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

7^n = (n-th row terms) dot (first n+1 odd integers). Example: 7^3 = 343 = (77, 63, 14, 1) dot (1, 3, 5, 7) = (77 + 189 + 70 + 7) = 243. - Gary W. Adamson, Jun 15 2011

			Triangle begins:
      1;
      4,     1;
     17,     9,    1;
     77,    63,   14,    1;
    371,   406,  134,   19,   1;
   1890,  2535, 1095,  230,  24,  1;
  10095, 15660, 8240, 2269, 351, 29, 1;
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  4, 1
  1, 5, 1
  0, 1, 5, 1
  0, 0, 1, 5, 1
  0, 0, 0, 1, 5, 1,
  0, 0, 0, 0, 1, 5, 1
  0, 0, 0, 0, 0, 1, 5, 1
  0, 0, 0, 0, 0, 0, 1, 5, 1
  0, 0, 0, 0, 0, 0, 0, 1, 5, 1 (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 4, 5], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Sum_{k=0..n} T(n,k) = A098409(n).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A104455(m+n).
Sum_{k=0..n} T(n,k)*(2*k+1) = 7^n. - Philippe Deléham, Mar 26 2007

A126791 Binomial matrix applied to A111418.

Original entry on oeis.org

1, 4, 1, 17, 7, 1, 75, 39, 10, 1, 339, 202, 70, 13, 1, 1558, 1015, 425, 110, 16, 1, 7247, 5028, 2400, 771, 159, 19, 1, 34016, 24731, 12999, 4872, 1267, 217, 22, 1, 160795, 121208, 68600, 28882, 8890, 1940, 284, 25, 1, 764388, 593019, 355890, 164136

Offset: 0

Philippe Deléham, Mar 14 2007

Triangle T(n,k), 0 <= k <= n, read by rows defined 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) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1.
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
From R. J. Mathar, Mar 12 2013: (Start)
The matrix inverse starts
      1;
     -4,     1;
     11,    -7,     1;
    -29,    31,   -10,    1;
     76,  -115,    60,  -13,    1;
   -199,   390,  -285,   98,  -16,   1;
    521, -1254,  1185, -566,  145, -19,   1;
  -1364,  3893, -4524, 2785, -985, 201, -22, 1; ... (End)

			Triangle begins:
      1;
      4,     1;
     17,     7,     1;
     75,    39,    10,    1;
    339,   202,    70,   13,    1;
   1558,  1015,   425,  110,   16,   1;
   7247,  5028,  2400,  771,  159,  19,  1;
  34016, 24731, 12999, 4872, 1267, 217, 22, 1; ...
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins:
  4, 1
  1, 3, 1
  0, 1, 3, 1
  0, 0, 1, 3, 1
  0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 1, 3, 1
  0, 0, 0, 0, 0, 0, 0, 1, 3, 1 (End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

A126791 := proc(n,k)
    if n=0 and k = 0 then
        1 ;
    elif k <0 or k>n then
        0;
    elif k= 0 then
        4*procname(n-1,0)+procname(n-1,1) ;
    else
        procname(n-1,k-1)+3*procname(n-1,k)+procname(n-1,k+1) ;
    end if;
end proc: # R. J. Mathar, Mar 12 2013
T := (n,k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1, -n+1, 3/2)): seq(seq(T(n,k),k=1..n),n=1..10); # Peter Luschny, May 13 2016

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0,
T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 4, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A026378(m+n+1).
Sum_{k=0..n} T(n,k) = 5^n = A000351(n).
T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1,-n+1,3/2)). - Peter Luschny, May 13 2016
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 + x )*(1 + 3*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

A126953 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) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

Original entry on oeis.org

1, 3, 1, 10, 3, 1, 33, 11, 3, 1, 110, 36, 12, 3, 1, 366, 122, 39, 13, 3, 1, 1220, 405, 135, 42, 14, 3, 1, 4065, 1355, 447, 149, 45, 15, 3, 1, 13550, 4512, 1504, 492, 164, 48, 16, 3, 1, 45162, 15054, 5004, 1668, 540, 180, 51, 17, 3, 1

Offset: 0

Philippe Deléham, Mar 19 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

Riordan array (2/(1-6x+sqrt(1-4*x^2)),x*c(x^2)) where c(x)= g.f. of the Catalan numbers A000108. - Philippe Deléham, Jun 01 2013

			Triangle begins:
     1;
     3,    1;
    10,    3,   1;
    33,   11,   3,   1;
   110,   36,  12,   3,  1;
   366,  122,  39,  13,  3,  1;
  1220,  405, 135,  42, 14,  3, 1;
  4065, 1355, 447, 149, 45, 15, 3, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];
Table[T[n, k, 3, 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

Sum_{k=0..n} T(n,k) = A127359(n).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A126931(m+n).
Sum_{k=0..n} T(n,k)*(-2*k+1) = 2^n. - Philippe Deléham, Mar 25 2007

A005775 Number of compact-rooted directed animals of size n having 3 source points.

Original entry on oeis.org

1, 4, 14, 45, 140, 427, 1288, 3858, 11505, 34210, 101530, 300950, 891345, 2638650, 7809000, 23107488, 68375547, 202336092, 598817490, 1772479905, 5247421410, 15538054455, 46019183840, 136325212750, 403933918375, 1197131976846, 3548715207534, 10521965227669

Offset: 3

Simon Plouffe

Binomial transform of A037955. - Paul Barry, Dec 28 2006
Apparently, the number of Dyck paths of semilength n that contain at least one UUU but avoid UUU's starting above level 0. - David Scambler, Jul 02 2013
a(n) = number of paths in the half-plane x >= 0 from (0,0) to (n-1,2) or (n-1,-3), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=5, we have the 14 paths: HHUU, UUHH, UHHU, HUUH, HUHU, UHUH, UDUU, UUDU, UUUD, DUUU, DDDH, HDDD, DHDD, DDHD. - José Luis Ramírez Ramírez, Apr 19 2015

			G.f. = x^3 + 4*x^4 + 14*x^5 + 45*x^6 + 140*x^7 + 427*x^8 + 1288*x^9 + 3858*x^10 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adv. in Appl. Math. 9 (1988), no. 3, 334-357.

Cf. A005773.
k=2 column of array in A038622.
Cf. A005774, A066822.

a005775 = flip a038622 2 . (subtract 1)  -- Reinhard Zumkeller, Feb 26 2013

seq(simplify(GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2)),n=3..28); # Peter Luschny, May 12 2016

nmax = 28; t[n_ /; n > 0, k_ /; k >= 1] := t[n, k] = t[n-1, k-1] + t[n-1, k] + t[n-1, k+1]; t[0, 0] = 1; t[0, ] = 0; t[?Negative, ?Negative] = 0; t[n, 0] := 2*t[n-1, 0] + t[n-1, 1]; a[n_] := t[n-1, 2]; Table[a[n], {n, 3, nmax} ] (* Jean-François Alcover, Jul 03 2013, from A038622 *)

{a(n) = polcoeff( (x^2 + x - 1 + (x^2 - 3*x + 1) * sqrt((1 + x) / (1 - 3*x) + x^3 * O(x^n))) / (2*x^2), n)};

{a(n) = n--; sum(k=0, n, binomial(n, k) * binomial(k, k\2 -1))}; /* Michael Somos, May 12 2016 */

D-finite with recurrence (n+2)*(n-3)*a(n) = 2*n*(n-1)*a(n-1) + 3*(n-1)*(n-2)*a(n-2), a(2)=0, a(3)=1. - Michael Somos, Feb 02 2002
G.f.: (x^2 + x - 1 +(x^2 - 3*x + 1)*sqrt((1+x)/(1-3*x)))/(2*x^2).
From Paul Barry, Dec 28 2006: (Start)
E.g.f.: exp(x)*(Bessel_I(2,2*x) + Bessel_I(3,2*x));
a(n+1) = Sum_{k=0..n} C(n,k)*C(k,floor(k/2)-1). (End)
a(n) ~ 3^(n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Feb 25 2014
G.f.: (z^3*M(z)^2+z^4*M(z)^3)/(1-z-2*z^2*M(z)), where M(z) is the g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 19 2015
a(n) = GegenbauerC(n-4,-n+1,-1/2) + GegenbauerC(n-3,-n+1,-1/2). - Peter Luschny, May 12 2016
0 = a(n)*(+9*a(n+1) - 63*a(n+2) - 54*a(n+3) + 87*a(n+4) - 21*a(n+5))+ a(n+1)*(+21*a(n+1) + 79*a(n+2) + 13*a(n+3) - 118*a(n+4) + 35*a(n+5)) + a(n+2)*(-14*a(n+2) + 79*a(n+3) - 67*a(n+4) + 14*a(n+5)) + a(n+3)*(+6*a(n+3) + 19*a(n+4) - 11*a(n+5)) + a(n+4)*(+a(n+4) + a(n+5)) if n >= 0. - Michael Somos, May 12 2016
a(n) = A005773(n) - A001006(n) for n >= 3. - John Keith, Nov 20 2020

More terms from Randall L Rathbun, Jan 19 2002

Edited by Michael Somos, Feb 02 2002

A333650 Triangle read by rows: T(n,k) gives the number of domino towers of height k consisting of n bricks.

Original entry on oeis.org

1, 1, 2, 1, 4, 4, 1, 7, 11, 8, 1, 12, 24, 28, 16, 1, 20, 52, 70, 68, 32, 1, 33, 110, 168, 193, 160, 64, 1, 54, 228, 401, 497, 510, 368, 128, 1, 88, 467, 944, 1257, 1412, 1304, 832, 256, 1, 143, 949, 2187, 3172, 3736, 3879, 3248, 1856, 512

Offset: 1

Peter Kagey, Mar 31 2020

The towers must have a contiguous base of bricks, and each brick must be at least half supported below by another brick. The stacks do not need to be stable.
Conjecture: For n > 1, T(n,2) = A000071(n+2).
A038622(n-1,k) appears to give the number of domino towers consisting of n bricks with a base of k bricks.
Conjecture: T(n,n-1) = A339252(n-2). - Peter Kagey, Nov 21 2020
Conjecture: T(n,n-2) = A339254(n-3). - Peter Luschny, Nov 29 2020
Conjecture: T(n,n-3) = A339029(n-4). - Peter Luschny, Dec 01 2020
From Peter Luschny, Dec 01 2020: (Start)
The above conjectures can be summarized as follows:
T(2*n + k, n + k) = d_{n}(n + k - 1) for k >= 1 and 0 <= n <= 3, where
d_{0}(m) = 2^(m-1)*2;
d_{1}(m) = 2^(m-3)*(10 + 6*m);
d_{2}(m) = 2^(m-5)*(70 + 43*m + 9*m^2);
d_{3}(m) = 2^(m-7)*(588 + 367*m + 84*m^2 + 9*m^3). (End)

			Table begins:
  n\k| 1   2    3    4    5    6     7     8    9   10   11
  ---+-----------------------------------------------------
   1 | 1
   2 | 1   2
   3 | 1   4    4
   4 | 1   7   11    8
   5 | 1  12   24   28   16
   6 | 1  20   52   70   68   32
   7 | 1  33  110  168  193  160    64
   8 | 1  54  228  401  497  510   368   128
   9 | 1  88  467  944 1257 1412  1304   832  256
  10 | 1 143  949 2187 3172 3736  3879  3248 1856  512
  11 | 1 232 1916 5010 7946 9778 10766 10360 7920 4096 1024
.
T(3,2) = 4 because there are four domino towers of height two consisting of three bricks:
+-------+-------+      +-------+                  +-------+
|       |       |      |       |                  |       |
+---+---+---+---+, +---+---+---+---+, +-------+---+---+---+, and
    |       |      |       |       |  |       |       |
    +-------+      +-------+-------+  +-------+-------+
+-------+
|       |
+---+---+---+-------+.
    |       |       |
    +-------+-------+

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 25-27.

Peter Luschny, Table of n, a(n), for row(k) for k = 1..18 (the first 14 rows by Peter Kagey).
J. Bétréma and J.-G. Penaud, Animaux et arbres guingois, Theoretical Computer Science 117, 67-89, 1993.
D. Gouyou-Beauchamps and G. Viennot, Equivalence of the two dimensional directed animals problem to a one-dimensional path problem, Adv. in Appl. Math. 9(3), 334-357, 1988.
Peter Kagey, Symmetric Brick Stacking, Mathematics Stack Exchange, 2018.
Doron Zeilberger, The amazing 3^n theorem and its even more amazing proof, arXiv:1208.2258 [math.CO], 2012.
Doron Zeilberger, The 27 towers with 4 domino pieces, illustration.

Cf. A000071 (col. 2), A339493 (col. 3), A000244, A038622, A168368, A264746, A320314, A339252, A339254, A339029, A339346, A339494.

Row sums are given by A000244(n-1) = 3^(n-1).
T(n,1) = 1.
T(n,n) = 2^(n-1).

A301475 Triangular array of polynomials related to the Motzkin triangle and to rooted polyominoes, coefficients in ascending order, read by rows, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 4, 5, 3, 1, 5, 3, 1, 3, 1, 1, 9, 12, 9, 4, 1, 12, 9, 4, 1, 9, 4, 1, 4, 1, 1, 21, 30, 25, 14, 5, 1, 30, 25, 14, 5, 1, 25, 14, 5, 1, 14, 5, 1, 5, 1, 1, 51, 76, 69, 44, 20, 6, 1, 76, 69, 44, 20, 6, 1, 69, 44, 20, 6, 1, 44, 20, 6, 1, 20, 6, 1, 6, 1, 1

Offset: 0

Peter Luschny, Mar 22 2018

Evaluating this triangle of polynomials at different values of x leads to interesting integer triangles. For instance at x = 0 it gives the Motzkin triangle A064189 (A026300), at x = 1 it counts rooted polyominoes A038622; at x = 2 it gives A126954 and at x =-1 gives A089942; x = 1/2 and scaling gives A301477.

			Triangle of polynomials starts:
                                    1
                                 1 + x, 1
                          2 + 2 x + x^2, 2 + x, 1
               4 + 5 x + 3 x^2  + x^3, 5 + 3 x^2 + x, 3 + x, 1
9 + 12 x + 9 x^2  + 4 x^3  + x^4, 12 + 9 x + 4 x^2 + x^3, 9 + 4 x + x^2, 4 + x, 1
.
Triangle of coefficients starts:
                               1
                            1, 1, 1
                        2, 2, 1, 2, 1, 1
                  4, 5, 3, 1, 5, 3, 1, 3, 1, 1
         9, 12, 9, 4, 1, 12, 9, 4, 1, 9, 4, 1, 4, 1, 1
21, 30, 25, 14, 5, 1, 30, 25, 14, 5, 1, 25, 14, 5, 1, 14, 5, 1, 5, 1, 1

Cf. A064189 (A026300) (x=0), A038622 (x=1), A126954 (x=2), A089942 (x=-1), A301477 (x=1/2, scaled).

Cf. A000244 (row sums), A000217 (row length).

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
T := (n,k) -> binomial(n,k)*hypergeom([-k/2,1/2-k/2], [-k+n+2], 4);
P := (n,m) -> add(simplify(T(n,k)*x^(n-k-m)), k=0..n-m);
for n from 0 to 5 do seq(sort(P(n,j),x,ascending), j=0..n) od;
for n from 0 to 5 do seq(CoeffList(P(n,j)), j=0..n) od;

P(n,k) = Sum_{j=0..n-k}binomial(n,j)*hypergeom([-j/2,1/2-j/2],[n-j+2],4)*x^(n-j-k).
T(n,k) is the list of the coefficients of P(n,k) in ascending order.
Row sums are powers of 3, row lengths are the triangular numbers.

cp's OEIS Frontend

A124733 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 (2,3,3,...) and super- and subdiagonals (1,1,1,...).

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A124576 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 (1,4,4,...) and super- and subdiagonals (1,1,1,...).

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A126970 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) = T(n-1,1), T(n,k) = T(n-1,k-1) + 3*T(n-1,k) + T(n-1,k+1) for k >= 1.

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A124574 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 (3,4,4,...) and super- and subdiagonals (1,1,1,...).

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A126331 Triangle T(n,k), 0 <= k <= n, read by rows defined 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) + 5*T(n-1,k) + T(n-1,k+1) for k >= 1.

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A126791 Binomial matrix applied to A111418.

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A126953 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) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

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A126331 Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 4T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + 5T(n-1,k) + T(n-1,k+1) for k >= 1.