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 A124733 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 (2,3,3,...) and super- and subdiagonals (1,1,1,...). 27
 1, 2, 1, 5, 5, 1, 15, 21, 8, 1, 51, 86, 46, 11, 1, 188, 355, 235, 80, 14, 1, 731, 1488, 1140, 489, 123, 17, 1, 2950, 6335, 5397, 2730, 875, 175, 20, 1, 12235, 27352, 25256, 14462, 5530, 1420, 236, 23, 1, 51822, 119547, 117582, 74172, 32472, 10026, 2151, 306, 26, 1 (list; table; graph; refs; listen; history; text; internal format)
 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 Equals A007318*A039599 (when written as lower triangular matrix). - Philippe Deléham, Jun 16 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, 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<=k<=n} (-1)^(n-k)*T(n,k) = (-1)^n. - Philippe Deléham, Feb 27 2007 Sum_{k, 0<=k<=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 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 Cf. A110877, A091965, A007317, A007317, A026375 (row sums). Sequence in context: A060920 A107842 A126216 * A137597 A059340 A248727 Adjacent sequences:  A124730 A124731 A124732 * A124734 A124735 A124736 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 October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)