A064189 -id:A064189 - cp's OEIS frontend

A125906 Riordan array (1/(1 + 5x + x^2), x/(1 + 5x + x^2))^(-1); inverse of Riordan array A123967.

1, 5, 1, 26, 10, 1, 140, 77, 15, 1, 777, 540, 153, 20, 1, 4425, 3630, 1325, 254, 25, 1, 25755, 23900, 10509, 2620, 380, 30, 1, 152675, 155764, 79065, 23989, 4550, 531, 35, 1, 919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1

Offset: 0

Philippe Deléham, Feb 04 2007

T(0)=A053121, T(1)=A064189, T(2)=A039598, T(3)=A091965, T(4)=A052179.
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 five types of steps H=(1,0); example: T(3,1)=77 because we have UDU, UUD, 25 HHU paths, 25 HUH paths and 25 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 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 terms in 1,2,3,...). Example: 7^3 = 343 = (140, 77, 15, 1) dot (1, 2, 3, 4) = (140 + 154 + 45 + 4) = 343. - Gary W. Adamson, Jun 17 2011
A subset of the "family of triangles" (Deleham 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
Riordan array (f(x), x*f(x)) where f(x) is the o.g.f. of A182401. - Philippe Deléham, Mar 04 2013

			Triangle begins
       1;
       5,       1;
      26,      10,      1;
     140,      77,     15,      1;
     777,     540,    153,     20,     1;
    4425,    3630,   1325,    254,    25,    1;
   25755,   23900,  10509,   2620,   380,   30,   1;
  152675,  155764,  79065,  23989,  4550,  531,  35,  1;
  919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1;
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins
  5, 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

Cf. A182401.

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, 5, 5], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Triangle T(5) where T(x) is defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1). Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0). Sum_{k=0..n} T(n,k) = A122898(n).
Sum_{k=0..n} T(n,k)*(k+1) = 7^n. - Philippe Deléham, Mar 26 2007
T(n,0) = A182401(n). - Philippe Deléham, Mar 04 2013
The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + 5*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

A126093 Inverse binomial matrix applied to A110877.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 6, 4, 1, 6, 18, 15, 6, 1, 18, 57, 54, 28, 8, 1, 57, 186, 193, 118, 45, 10, 1, 186, 622, 690, 474, 218, 66, 12, 1, 622, 2120, 2476, 1856, 976, 362, 91, 14, 1, 2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1

Offset: 0

Philippe Deléham, Mar 03 2007

Diagonal sums are A065601. - Philippe Deléham, Mar 05 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

			Triangle begins:
     1;
     0,    1;
     1,    2,    1;
     2,    6,    4,    1;
     6,   18,   15,    6,    1;
    18,   57,   54,   28,    8,    1;
    57,  186,  193,  118,   45,   10,   1;
   186,  622,  690,  474,  218,   66,  12,   1;
   622, 2120, 2476, 1856,  976,  362,  91,  14,  1;
  2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1;
Production matrix begins
  0, 1;
  1, 2, 1;
  0, 1, 2, 1;
  0, 0, 1, 2, 1;
  0, 0, 0, 1, 2, 1;
  0, 0, 0, 0, 1, 2, 1;
  0, 0, 0, 0, 0, 1, 2, 1;
  0, 0, 0, 0, 0, 0, 1, 2, 1;
  0, 0, 0, 0, 0, 0, 0, 1, 2, 1;
- _Philippe Deléham_, Nov 07 2011

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths, Eur. J. Comb. 39, 157-169 (2014), Table 2.2.

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,0,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 21 2017 *)

@CachedFunction
def T(n, k, x, y):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    elif (k==0): return x*T(n-1,0,x,y) + T(n-1,1,x,y)
    else: return T(n-1,k-1,x,y) + y*T(n-1,k,x,y) + T(n-1,k+1,x,y)
[[T(n,k,0,2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020

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) = T(n-1,1), T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k>=1.
Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0) = A000957(m+n+1).
Sum_{k=0..n-1} T(n,k) = A026641(n), for n>=1. - Philippe Deléham, Mar 05 2007
Sum_{k=0..n} T(n,k)*(3k+1) = 4^n. - Philippe Deléham, Mar 22 2007

A126954 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) + T(n-1,k+1) for k >= 1.

Original entry on oeis.org

1, 3, 1, 10, 4, 1, 34, 15, 5, 1, 117, 54, 21, 6, 1, 405, 192, 81, 28, 7, 1, 1407, 678, 301, 116, 36, 8, 1, 4899, 2386, 1095, 453, 160, 45, 9, 1, 17083, 8380, 3934, 1708, 658, 214, 55, 10, 1, 59629, 29397, 14022, 6300, 2580, 927, 279, 66, 11, 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;
     3,    1;
    10,    4,    1;
    34,   15,    5,   1;
   117,   54,   21,   6,   1;
   405,  192,   81,  28,   7,  1;
  1407,  678,  301, 116,  36,  8, 1;
  4899, 2386, 1095, 453, 160, 45, 9, 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, 1], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

Sum_{k=0..n} T(n,k) = A126932(n).
Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A059738(m+n).
Sum_{k=0..n} T(n,k)*(-k+1) = 3^n. - Philippe Deléham, Mar 26 2007

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,...).

Original entry on oeis.org

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

cp's OEIS Frontend

A125906 Riordan array (1/(1 + 5*x + x^2), x/(1 + 5*x + x^2))^(-1); inverse of Riordan array A123967.

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A126093 Inverse binomial matrix applied to A110877.

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A126954 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) + T(n-1,k+1) for k >= 1.

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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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Mathematica

A125906 Riordan array (1/(1 + 5x + x^2), x/(1 + 5x + x^2))^(-1); inverse of Riordan array A123967.

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.