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%I #31 Sep 07 2022 09:50:15
%S 1,5,1,26,10,1,140,77,15,1,777,540,153,20,1,4425,3630,1325,254,25,1,
%T 25755,23900,10509,2620,380,30,1,152675,155764,79065,23989,4550,531,
%U 35,1,919139,1010560,575078,203560,47270,7240,707,40,1
%N Riordan array (1/(1 + 5*x + x^2), x/(1 + 5*x + x^2))^(-1); inverse of Riordan array A123967.
%C T(0)=A053121, T(1)=A064189, T(2)=A039598, T(3)=A091965, T(4)=A052179.
%C Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and five types of steps H=(1,0); example: T(3,1)=77 because we have UDU, UUD, 25 HHU paths, 25 HUH paths and 25 UHH paths. - _Philippe Deléham_, Sep 25 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 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
%C 7^n = (n-th row terms) dot (first n+1 terms in 1,2,3,...). Example: 7^3 = 343 = (140, 77, 15, 1) dot (1, 2, 3, 4) = (140 + 154 + 45 + 4) = 343. - _Gary W. Adamson_, Jun 17 2011
%C A subset of the "family of triangles" (Deleham comment of Sep 25 2007) is the succession of binomial transforms beginning with triangle A053121, (0,0); giving -> A064189, (1,1); -> A039598, (2,2); -> A091965, (3,3); -> A052179, (4,4); -> A125906, (5,5) ->, etc; generally the binomial transform of the triangle generated from (n,n) = that generated from ((n+1),(n+1)). - _Gary W. Adamson_, Aug 03 2011
%C Riordan array (f(x), x*f(x)) where f(x) is the o.g.f. of A182401. - _Philippe Deléham_, Mar 04 2013
%H G. C. Greubel, <a href="/A125906/b125906.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F Triangle T(5) where T(x) is defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1). Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0). Sum_{k=0..n} T(n,k) = A122898(n).
%F Sum_{k=0..n} T(n,k)*(k+1) = 7^n. - _Philippe Deléham_, Mar 26 2007
%F T(n,0) = A182401(n). - _Philippe Deléham_, Mar 04 2013
%F The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + 5*x + x^2)^n expanded about the point x = 0. - _Peter Bala_, Sep 06 2022
%e Triangle begins
%e 1;
%e 5, 1;
%e 26, 10, 1;
%e 140, 77, 15, 1;
%e 777, 540, 153, 20, 1;
%e 4425, 3630, 1325, 254, 25, 1;
%e 25755, 23900, 10509, 2620, 380, 30, 1;
%e 152675, 155764, 79065, 23989, 4550, 531, 35, 1;
%e 919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1;
%e From _Philippe Deléham_, Nov 07 2011: (Start)
%e Production matrix begins
%e 5, 1;
%e 1, 5, 1,;
%e 0, 1, 5, 1;
%e 0, 0, 1, 5, 1;
%e 0, 0, 0, 1, 5, 1;
%e 0, 0, 0, 0, 1, 5, 1;
%e 0, 0, 0, 0, 0, 1, 5, 1;
%e 0, 0, 0, 0, 0, 0, 1, 5, 1;
%e 0, 0, 0, 0, 0, 0, 0, 1, 5, 1; (End)
%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, 5, 5], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, May 22 2017 *)
%Y Cf. A182401.
%K nonn,tabl
%O 0,2
%A _Philippe Deléham_, Feb 04 2007
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