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%I #40 Sep 07 2022 09:51:00
%S 1,3,1,10,7,1,37,39,11,1,150,204,84,15,1,654,1050,555,145,19,1,3012,
%T 5409,3415,1154,222,23,1,14445,28063,20223,8253,2065,315,27,1,71398,
%U 146920,117208,55300,16828,3352,424,31,1,361114,776286,671052,355236,125964,30660,5079,549,35,1
%N 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,...).
%C Column 1 yields A064613. Row sums yield A081671.
%C 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
%C 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
%C 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
%C 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
%C From _Peter Bala_, Sep 06 2022: (Start)
%C The following assume the row and column indexing start at 0.
%C 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.
%C 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.
%C 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)
%H G. C. Greubel, <a href="/A124574/b124574.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F Sum_{k=0..n} (-1)^(n-k)*T(n,k) = (-2)^n. - _Philippe Deléham_, Feb 27 2007
%F Sum_{k=0..n} T(n,k)*(2*k+1) = 6^n. - _Philippe Deléham_, Mar 27 2007
%F T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,2) + GegenbauerC(n-k-1,-n+1,2)). - _Peter Luschny_, May 13 2016
%e 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].
%e Triangle starts:
%e 1;
%e 3, 1
%e 10, 7, 1;
%e 37, 39, 11, 1
%e 150, 204, 84, 15, 1;
%e 654, 1050, 555, 145, 19, 1;
%e From _Philippe Deléham_, Nov 07 2011: (Start)
%e Production matrix begins:
%e 3, 1
%e 1, 4, 1
%e 0, 1, 4, 1
%e 0, 0, 1, 4, 1
%e 0, 0, 0, 1, 4, 1
%e 0, 0, 0, 0, 1, 4, 1
%e 0, 0, 0, 0, 0, 1, 4, 1
%e 0, 0, 0, 0, 0, 0, 1, 4, 1
%e 0, 0, 0, 0, 0, 0, 0, 1, 4, 1 (End)
%p 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
%p 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
%t 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 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 *)
%Y Cf. A000108, A081671 (row sums), A124575, A124576, A052179, A064613, A005572.
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_ & _Roger L. Bagula_, Nov 04 2006
%E Edited by _N. J. A. Sloane_, Dec 04 2006
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