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 A126093 Inverse binomial matrix applied to A110877. 28
 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, 1, 186, 622, 690, 474, 218, 66, 12, 1, 622, 2120, 2476, 1856, 976, 362, 91, 14, 1, 2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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. Diagonal sums are A065601. - Philippe Deléham, Mar 05 2007 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 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened Sun, Yidong; Ma, Luping Minors of a class of Riordan arrays related to weighted partial Motzkin paths.  Eur. J. Comb. 39, 157-169 (2014), Table 2.2. FORMULA Sum_{k, 0<=k<=n} T(m,k)*T(n,k) = T(m+n,0) = A000957(m+n+1). Sum_{k, 0<=k<=n-1} T(n,k) = A026641(n), for n>=1. - Philippe Deléham, Mar 05 2007 Sum_{k, 0<=k<=n} T(n,k)*(3k+1) = 4^n. - Philippe Deléham, Mar 22 2007 EXAMPLE Triangle begins: 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, 1; 186, 622, 690, 474, 218, 66, 12, 1; 622, 2120, 2476, 1856, 976, 362, 91, 14, 1; 2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1; Production matrix begins 0, 1 1, 2, 1 0, 1, 2, 1 0, 0, 1, 2, 1 0, 0, 0, 1, 2,1 0, 0, 0, 0, 1, 2, 1 0, 0, 0, 0, 0, 1, 2, 1 0, 0, 0, 0, 0, 0, 1, 2, 1 0, 0, 0, 0, 0, 0, 0, 1, 2, 1 - Philippe Deléham, Nov 07 2011 MATHEMATICA 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, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *) CROSSREFS Sequence in context: A121341 A241737 A174959 * A065279 A151962 A072137 Adjacent sequences:  A126090 A126091 A126092 * A126094 A126095 A126096 KEYWORD nonn,tabl AUTHOR Philippe Deléham, Mar 03 2007 STATUS approved

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)