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

Author

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

Keywords

• nonn
• tabl

Comments

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]

Examples

			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;
		

Links

• 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.

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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)

Extensions

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