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 A124575 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 (2,4,4,...) and super- and subdiagonals (1,1,1,...). 29
 1, 2, 1, 5, 6, 1, 16, 30, 10, 1, 62, 146, 71, 14, 1, 270, 717, 444, 128, 18, 1, 1257, 3582, 2621, 974, 201, 22, 1, 6096, 18206, 15040, 6718, 1800, 290, 26, 1, 30398, 93960, 85084, 43712, 14208, 2986, 395, 30, 1, 154756, 491322, 478008, 274140, 103530 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Column k=0 yields A033543 (2nd binomial transform of the sequence A000957(n+1)). Row sums yield A133158. [Corrected by Philippe Deléham, Oct 24 2007, Dec 05 2009] 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) + 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 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 LINKS G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened FORMULA T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + T(n-1,k-1) for k >= 2. Sum_{k=0..n} T(n,k)*(3*k+1) = 6^n. - Philippe Deléham, Mar 27 2007 Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A033543(m+n). - Philippe Deléham, Nov 22 2009 EXAMPLE Row 2 is (5,6,1) because M= [2,1,0;1,4,1;0,1,4] and M^2=[5,6,1;6,18,8;1,8,17]. Triangle starts:     1;     2,   1;     5,   6,   1;    16,  30,  10,   1;    62, 146,  71,  14,  1;   270, 717, 444, 128, 18, 1; MAPLE with(linalg): m:=proc(i, j) if i=1 and j=1 then 2 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; 2, 1; for n from 3 to 11 do seq(B[n][1, j], j=1..n) od; # yields sequence in triangular form MATHEMATICA M[n_] := SparseArray[{{1, 1} -> 2, Band[{2, 2}] -> 4, Band[{1, 2}] -> 1, Band[{2, 1}] -> 1}, {n, n}]; row = {1}; row[n_] := MatrixPower[M[n], n-1] // First // Normal; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 09 2014 *) CROSSREFS Cf. A124576, A124574, A052179, A064613, A133158. Sequence in context: A179455 A039810 A328297 * A178121 A302595 A113345 Adjacent sequences:  A124572 A124573 A124574 * A124576 A124577 A124578 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

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Last modified May 7 19:23 EDT 2021. Contains 343652 sequences. (Running on oeis4.)