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A126331
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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)=4*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+5*T(n-1,k)+T(n-1,k+1) for k>=1.
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25
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1, 4, 1, 17, 9, 1, 77, 63, 14, 1, 371, 406, 134, 19, 1, 1890, 2535, 1095, 230, 24, 1, 10095, 15660, 8240, 2269, 353, 29, 1, 56040, 96635, 59129, 19936, 4053, 497, 34, 1, 320795, 598344, 512216, 162862, 40698, 6572, 668, 39, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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 odd integers). Example: 7^3 = 343 = (77, 63, 14, 1) dot (1, 3, 5, 7) = (77 + 189 + 70 + 7) = 243. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2011
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FORMULA
| Sum_{k, 0<=k<=n}T(n,k)=A098409(n). Sum_{k, k>=0}T(m,k)*T(n,k)=T(m+n,0)=A104455(m+n).
Sum_{k, 0<=k<=n}T(n,k)*(2*k+1)=7^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 26 2007
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EXAMPLE
| Triangle begins:
1;
4, 1;
17, 9, 1;
77, 63, 14, 1;
371, 406, 134, 19, 1;
1890, 2535, 1095, 230, 24, 1;
10095, 15660, 8240, 2269, 351, 29, 1;
Production matrix begins
4, 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: A126791 A052179 A171589 * A013631 A113355 A201201
Adjacent sequences: A126328 A126329 A126330 * A126332 A126333 A126334
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 10 2007
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