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1, 1, 1, 1, 4, 1, 1, 8, 8, 1, 1, 13, 25, 13, 1, 1, 19, 59, 59, 19, 1, 1, 26, 119, 194, 119, 26, 1, 1, 34, 216, 524, 524, 216, 34, 1, 1, 43, 363, 1231, 1833, 1231, 363, 43, 1, 1, 53, 575, 2603, 5417, 5417, 2603, 575, 53, 1, 1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Row sums = A132790: (1, 2, 6, 18, 53, 158,...). Column next to left 1's border = A034856: (1, 4, 8, 13, 19, 26,...).
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FORMULA
| A007318 + A001263 - A000012 as infinite lower triangular matrices.
A symmetrical triangle recursion: let q=4; t(n,m,0)=Binomial[n,m]; t(n,m,1)=Narayana(n,m); t(n,m,2)=Eulerian(n+1,m); t(n,m,q)=t(n,m,g-2)+t(n,m,q-3)
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EXAMPLE
| First few rows of the triangle are:
{1},
{1, 1},
{1, 4, 1},
{1, 8, 8, 1},
{1, 13, 25, 13, 1},
{1, 19, 59, 59, 19, 1},
{1, 26, 119, 194, 119, 26, 1},
{1, 34, 216, 524, 524, 216, 34, 1},
{1, 43, 363, 1231, 1833, 1231, 363, 43, 1},
{1, 53, 575, 2603, 5417, 5417, 2603, 575, 53, 1},
{1, 64, 869, 5069, 14069, 19655, 14069, 5069, 869, 64, 1}
...
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MATHEMATICA
| << DiscreteMath`Combinatorica`
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Binomial[n, m]*Binomial[n + 1, m]/(m + 1);
t[n_, m_, 2] := Eulerian[1 + n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
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CROSSREFS
| Cf. A007318, A001263, A034856, A132790.
Sequence in context: A158687 A141541 A177947 * A100754 A174035 A055107
Adjacent sequences: A132786 A132787 A132788 * A132790 A132791 A132792
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 30 2007
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EXTENSIONS
| More terms, Mma program and additional comments from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 20 2010
Edited by N. J. A. Sloane, Apr 21 2010 at the suggestion of R. J. Mathar
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