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Inverse binomial matrix applied to A110877.
28

%I #29 Sep 17 2024 20:55:26

%S 1,0,1,1,2,1,2,6,4,1,6,18,15,6,1,18,57,54,28,8,1,57,186,193,118,45,10,

%T 1,186,622,690,474,218,66,12,1,622,2120,2476,1856,976,362,91,14,1,

%U 2120,7338,8928,7164,4170,1791,558,120,16,1

%N Inverse binomial matrix applied to A110877.

%C Diagonal sums are A065601. - _Philippe Deléham_, Mar 05 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

%H G. C. Greubel, <a href="/A126093/b126093.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Yidong Sun and Luping Ma, <a href="https://doi.org/10.1016/j.ejc.2014.01.004">Minors of a class of Riordan arrays related to weighted partial Motzkin paths</a>, Eur. J. Comb. 39, 157-169 (2014), Table 2.2.

%F 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) = T(n-1,1), T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k>=1.

%F Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0) = A000957(m+n+1).

%F Sum_{k=0..n-1} T(n,k) = A026641(n), for n>=1. - _Philippe Deléham_, Mar 05 2007

%F Sum_{k=0..n} T(n,k)*(3k+1) = 4^n. - _Philippe Deléham_, Mar 22 2007

%e Triangle begins:

%e 1;

%e 0, 1;

%e 1, 2, 1;

%e 2, 6, 4, 1;

%e 6, 18, 15, 6, 1;

%e 18, 57, 54, 28, 8, 1;

%e 57, 186, 193, 118, 45, 10, 1;

%e 186, 622, 690, 474, 218, 66, 12, 1;

%e 622, 2120, 2476, 1856, 976, 362, 91, 14, 1;

%e 2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1;

%e Production matrix begins

%e 0, 1;

%e 1, 2, 1;

%e 0, 1, 2, 1;

%e 0, 0, 1, 2, 1;

%e 0, 0, 0, 1, 2, 1;

%e 0, 0, 0, 0, 1, 2, 1;

%e 0, 0, 0, 0, 0, 1, 2, 1;

%e 0, 0, 0, 0, 0, 0, 1, 2, 1;

%e 0, 0, 0, 0, 0, 0, 0, 1, 2, 1;

%e - _Philippe Deléham_, Nov 07 2011

%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,0,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 21 2017 *)

%o (Sage)

%o @CachedFunction

%o def T(n, k, x, y):

%o if (k<0 or k>n): return 0

%o elif (n==0 and k==0): return 1

%o elif (k==0): return x*T(n-1,0,x,y) + T(n-1,1,x,y)

%o else: return T(n-1,k-1,x,y) + y*T(n-1,k,x,y) + T(n-1,k+1,x,y)

%o [[T(n,k,0,2) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 27 2020

%K nonn,tabl

%O 0,5

%A _Philippe Deléham_, Mar 03 2007