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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 * A322383 A295856 A052964 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 December 15 21:22 EST 2018. Contains 318154 sequences. (Running on oeis4.)