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%I #28 Sep 07 2022 09:50:07
%S 1,4,1,17,7,1,75,39,10,1,339,202,70,13,1,1558,1015,425,110,16,1,7247,
%T 5028,2400,771,159,19,1,34016,24731,12999,4872,1267,217,22,1,160795,
%U 121208,68600,28882,8890,1940,284,25,1,764388,593019,355890,164136
%N Binomial matrix applied to A111418.
%C Triangle T(n,k), 0 <= k <= n, read by rows defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 4*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.
%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 From _R. J. Mathar_, Mar 12 2013: (Start)
%C The matrix inverse starts
%C 1;
%C -4, 1;
%C 11, -7, 1;
%C -29, 31, -10, 1;
%C 76, -115, 60, -13, 1;
%C -199, 390, -285, 98, -16, 1;
%C 521, -1254, 1185, -566, 145, -19, 1;
%C -1364, 3893, -4524, 2785, -985, 201, -22, 1; ... (End)
%H G. C. Greubel, <a href="/A126791/b126791.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A026378(m+n+1).
%F Sum_{k=0..n} T(n,k) = 5^n = A000351(n).
%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 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. - _Peter Bala_, Sep 06 2022
%e Triangle begins:
%e 1;
%e 4, 1;
%e 17, 7, 1;
%e 75, 39, 10, 1;
%e 339, 202, 70, 13, 1;
%e 1558, 1015, 425, 110, 16, 1;
%e 7247, 5028, 2400, 771, 159, 19, 1;
%e 34016, 24731, 12999, 4872, 1267, 217, 22, 1; ...
%e From _Philippe Deléham_, Nov 07 2011: (Start)
%e Production matrix begins:
%e 4, 1
%e 1, 3, 1
%e 0, 1, 3, 1
%e 0, 0, 1, 3, 1
%e 0, 0, 0, 1, 3, 1
%e 0, 0, 0, 0, 1, 3, 1
%e 0, 0, 0, 0, 0, 1, 3, 1
%e 0, 0, 0, 0, 0, 0, 1, 3, 1
%e 0, 0, 0, 0, 0, 0, 0, 1, 3, 1 (End)
%p A126791 := proc(n,k)
%p if n=0 and k = 0 then
%p 1 ;
%p elif k <0 or k>n then
%p 0;
%p elif k= 0 then
%p 4*procname(n-1,0)+procname(n-1,1) ;
%p else
%p procname(n-1,k-1)+3*procname(n-1,k)+procname(n-1,k+1) ;
%p end if;
%p end proc: # _R. J. Mathar_, Mar 12 2013
%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 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, 4, 3], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, May 22 2017 *)
%K nonn,tabl
%O 0,2
%A _Philippe Deléham_, Mar 14 2007
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