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A126093
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Inverse binomial matrix applied to A110877.
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28
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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)
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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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.
Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0) = A000957(m+n+1).
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(Sage)
@CachedFunction
def T(n, k, x, y):
if (k<0 or k>n): return 0
elif (n==0 and k==0): return 1
elif (k==0): return x*T(n-1, 0, x, y) + T(n-1, 1, x, y)
else: return T(n-1, k-1, x, y) + y*T(n-1, k, x, y) + T(n-1, k+1, x, y)
[[T(n, k, 0, 2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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