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%I #35 Sep 07 2022 09:50:49
%S 1,2,1,5,5,1,15,21,8,1,51,86,46,11,1,188,355,235,80,14,1,731,1488,
%T 1140,489,123,17,1,2950,6335,5397,2730,875,175,20,1,12235,27352,25256,
%U 14462,5530,1420,236,23,1,51822,119547,117582,74172,32472,10026,2151,306,26,1
%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 (2,3,3,...) and super- and subdiagonals (1,1,1,...).
%C 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
%C Equals A007318*A039599 (when written as lower triangular matrix). - _Philippe Deléham_, Jun 16 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 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
%C 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]
%H G. C. Greubel, <a href="/A124733/b124733.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Shu-Chiuan Chang and Robert Shrock, <a href="https://doi.org/10.1007/s10955-009-9868-0">Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips</a>, Journal of Statistical Physics 137 (2009) 667.
%F Sum_{k, 0<=k<=n} (-1)^(n-k)*T(n,k) = (-1)^n. - _Philippe Deléham_, Feb 27 2007
%F Sum_{k, 0<=k<=n} T(n,k)*(2*k+1) = 5^n. - _Philippe Deléham_, Mar 27 2007
%F 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
%F From _Peter Bala_, Sep 06 2022: (Start)
%F The following assume the row and column indexing start at 0.
%F Riordan array (f(x), x*g(x)), where f(x) = ( 1 - sqrt((1 - 5*x)/(1 - x)) )/(2*x) = 1 + 2*x + 5*x^2 + 15*x^3 + 51*x^4 + ... is the o.g.f. of A007317 and g(x) = ( 1 - 3*x - sqrt(1 - 6*x + 5*x^2) )/(2*x^2) = 1 + 3*x + 10*x^2 + 36*x^3 + 137*x^4 + .... See A002212.
%F The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x)*(1 + 3*x + x^2)^n expanded about the point x = 0.
%F 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)*3^(2*j+k-n). (End)
%e 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].
%e Triangle starts:
%e 1;
%e 2, 1;
%e 5, 5, 1;
%e 15, 21, 8, 1;
%e 51, 86, 46, 11, 1;
%e 188, 355, 235, 80, 14, 1;
%p 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
%p 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
%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]];
%t Table[T[n, k, 2, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Apr 21 2017 *)
%Y Cf. A110877, A091965, A002212, A007317, A026375 (row sums).
%K nonn,tabl
%O 1,2
%A _Gary W. Adamson_ & _Roger L. Bagula_, Nov 05 2006
%E Edited by _N. J. A. Sloane_, Dec 04 2006
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