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A126093 Inverse binomial matrix applied to A110877.

%I A126093 #29 Sep 17 2024 20:55:26
%S A126093 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 A126093 1,186,622,690,474,218,66,12,1,622,2120,2476,1856,976,362,91,14,1,
%U A126093 2120,7338,8928,7164,4170,1791,558,120,16,1
%N A126093 Inverse binomial matrix applied to A110877.
%C A126093 Diagonal sums are A065601. - _Philippe Deléham_, Mar 05 2007
%C A126093 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 A126093 G. C. Greubel, <a href="/A126093/b126093.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A126093 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 A126093 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 A126093 Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0) = A000957(m+n+1).
%F A126093 Sum_{k=0..n-1} T(n,k) = A026641(n), for n>=1. - _Philippe Deléham_, Mar 05 2007
%F A126093 Sum_{k=0..n} T(n,k)*(3k+1) = 4^n. - _Philippe Deléham_, Mar 22 2007
%e A126093 Triangle begins:
%e A126093      1;
%e A126093      0,    1;
%e A126093      1,    2,    1;
%e A126093      2,    6,    4,    1;
%e A126093      6,   18,   15,    6,    1;
%e A126093     18,   57,   54,   28,    8,    1;
%e A126093     57,  186,  193,  118,   45,   10,   1;
%e A126093    186,  622,  690,  474,  218,   66,  12,   1;
%e A126093    622, 2120, 2476, 1856,  976,  362,  91,  14,  1;
%e A126093   2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1;
%e A126093 Production matrix begins
%e A126093   0, 1;
%e A126093   1, 2, 1;
%e A126093   0, 1, 2, 1;
%e A126093   0, 0, 1, 2, 1;
%e A126093   0, 0, 0, 1, 2, 1;
%e A126093   0, 0, 0, 0, 1, 2, 1;
%e A126093   0, 0, 0, 0, 0, 1, 2, 1;
%e A126093   0, 0, 0, 0, 0, 0, 1, 2, 1;
%e A126093   0, 0, 0, 0, 0, 0, 0, 1, 2, 1;
%e A126093 - _Philippe Deléham_, Nov 07 2011
%t A126093 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 A126093 (Sage)
%o A126093 @CachedFunction
%o A126093 def T(n, k, x, y):
%o A126093     if (k<0 or k>n): return 0
%o A126093     elif (n==0 and k==0): return 1
%o A126093     elif (k==0): return x*T(n-1,0,x,y) + T(n-1,1,x,y)
%o A126093     else: return T(n-1,k-1,x,y) + y*T(n-1,k,x,y) + T(n-1,k+1,x,y)
%o A126093 [[T(n,k,0,2) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 27 2020
%K A126093 nonn,tabl
%O A126093 0,5
%A A126093 _Philippe Deléham_, Mar 03 2007