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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,...). 25
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column 1 yields A064613 (2nd binomial transform of the Catalan sequence A000108). 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 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

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

LINKS

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

FORMULA

Sum_{k, 0<=k<=n}(-1)^(n-k)*T(n,k)=(-2)^n . - Philippe Deléham, Feb 27 2007

Sum_{k, 0<=k<=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

EXAMPLE

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;

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

- From Philippe Deléham, Nov 07 2011

MAPLE

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

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A116384 A117207 A046658 * A295856 A052964 A084178

Adjacent sequences:  A124571 A124572 A124573 * A124575 A124576 A124577

KEYWORD

nonn,tabl

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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Last modified October 15 22:21 EDT 2018. Contains 316252 sequences. (Running on oeis4.)