OFFSET
1,2
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
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]
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.
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)
EXAMPLE
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;
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson & Roger L. Bagula, Nov 05 2006
EXTENSIONS
Edited by N. J. A. Sloane, Dec 04 2006
STATUS
approved