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%I #36 Feb 07 2022 02:42:25
%S 1,1,1,2,4,1,6,15,7,1,21,58,37,10,1,79,232,179,68,13,1,311,954,837,
%T 396,108,16,1,1265,4010,3861,2133,736,157,19,1,5275,17156,17726,10996,
%U 4498,1226,215,22,1,22431,74469,81330,55212,25716,8391,1893
%N Triangle T(n,k), 0 <= k <= n, read by rows, defined by: T(0,0) = 1, T(n,k) = 0 if n<k, T(n,0) = T(n-1,0) + T(n-1,1) and for k >= 1: T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1) with x = 3.
%C Similar to A064189 (x = 1) and to A039599 (x = 2).
%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 Row sums yield A126568. - _Philippe Deléham_, Oct 10 2007
%C 5^n = (n-th row terms) dot (first n+1 terms in the series (1, 4, 7, 10, ...)). Example for row 4: 5^4 = 625 = (21, 58, 37, 10, 1) dot (1, 4, 7, 10, 13) = (21 + 232 + 259 + 100 + 13). - _Gary W. Adamson_, Jun 15 2011
%C Riordan array (2/(1+x+sqrt(1-6*x+5*x^2)), (1-3*x-sqrt(1-6*x+5*x^2))/(2*x)). - _Philippe Deléham_, Mar 04 2013
%H G. C. Greubel, <a href="/A110877/b110877.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F T(n, 0) = A033321(n) and for k >= 1: T(n, k) = Sum_{j>=1} T(n-j, k-1)*A002212(j).
%F Sum_{k=0..n} T(m, k)*T(n, k) = T(m+n, 0) = A033321(m+n).
%F The triangle may also be generated from M^n * [1,0,0,0,...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and (1,3,3,3,...) in the main diagonal. - _Gary W. Adamson_, Dec 17 2006
%F Sum_{k=0..n} T(n,k)*(3*k+1) = 5^n. - _Philippe Deléham_, Feb 26 2007
%F Sum_{k=0..n} T(n,k) = A126568(n). - _Philippe Deléham_, Oct 10 2007
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 4, 1;
%e 6, 15, 7, 1;
%e 21, 58, 37, 10, 1;
%e 79, 232, 179, 68, 13, 1;
%e 311, 954, 837, 396, 108, 16, 1;
%e 1265, 4010, 3861, 2133, 736, 157, 19, 1;
%e 5275, 17156, 17726, 10996, 4498, 1226, 215, 22, 1;
%e 22431, 74469, 81330, 55212, 25716, 8391, 1893, 282, 25, 1;
%e ...
%e From _Philippe Deléham_, Nov 07 2011: (Start)
%e Production matrix begins:
%e 1, 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;
%e ... (End)
%p A110877 := proc(n,k)
%p if k > n then
%p 0;
%p elif n= 0 then
%p 1;
%p elif k = 0 then
%p 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_, Sep 06 2013
%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]]; Table[T[n, k, 1, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Apr 21 2017 *)
%Y Cf. A002212, A033321, A039599, A064189.
%Y The inverse of A126126.
%K nonn,easy,tabl
%O 0,4
%A _Philippe Deléham_, Sep 19 2005
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