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A125906
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Riordan array (1/(1+5*x+x^2),x/(1+5*x+x^2))^(-1); inverse of Riordan array A123967.
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26
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1, 5, 1, 26, 10, 1, 140, 77, 15, 1, 777, 540, 153, 20, 1, 4425, 3630, 1325, 254, 25, 1, 25755, 23900, 10509, 2620, 380, 30, 1, 152675, 155764, 79065, 23989, 4550, 531, 35, 1, 919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| T(0)= A053121, T(1)= A064189, T(2)= A039598, T(3)= A091965, T(4)= A052179.
Triangle read by rows:T(n,k)=number of lattice paths from (0,0) to (n,k)that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and five types of steps H=(1,0); example: T(3,1)=77 because we have UDU, UUD, 25 HHU paths, 25 HUH paths and 25 UHH paths . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 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 DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007
7^n = (n-th row terms) dot (first n+1 terms in 1,2,3,...). Example: 7^3 = 343 = (140, 77, 15, 1) dot (1, 2, 3, 4) = (140 + 154 + 45 + 4) = 343. - Gary W. Adamson, Jun 17 2011
A subset of the "family of triangles" (Deleham comment of Sep 25 2007) is the succession of binomial transforms beginning with triangle A053121, (0,0); giving -> A064189, (1,1); -> A039598, (2,2); -> A091965, (3,3); -> A052179, (4,4); -> A125906, (5,5) ->, etc; generally the binomial transform of the triangle generated from (n,n) = that generated from ((n+1),(n+1)). - Gary W. Adamson, Aug 03 2011
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FORMULA
| Triangle T(5)where T(x) is defined by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,k)=T(n-1,k-1)+x*T(n-1,k)+T(n-1,k+1) . Sum_{k, 0<=k<=n}T(m,k)*T(n,k)=T(m+n,0). Sum_{k, 0<=k<=n}T(n,k)=A122898(n).
Sum_{k, 0<=k<=n}T(n,k)*(k+1)=7^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 26 2007
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EXAMPLE
| Triangle begins:
1;
5, 1;
26, 10, 1;
140, 77, 15, 1;
777, 540, 153, 20, 1;
4425, 3630, 1325, 254, 25, 1;
25755, 23900, 10509, 2620, 380, 30, 1;
152675, 155764, 79065, 23989, 4550, 531, 35, 1;
919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1;
Production matrix begins
5, 1
1, 5, 1,
0, 1, 5, 1
0, 0, 1, 5, 1
0, 0, 0, 1, 5, 1
0, 0, 0, 0, 1, 5, 1
0, 0, 0, 0, 0, 1, 5, 1
0, 0, 0, 0, 0, 0, 1, 5, 1
0, 0, 0, 0, 0, 0, 0, 1, 5, 1
- From DELEHAM Philippe, Nov 07 2011
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CROSSREFS
| Sequence in context: A075500 A096645 A140713 * A146414 A146374 A188647
Adjacent sequences: A125903 A125904 A125905 * A125907 A125908 A125909
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 04 2007
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