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A132813
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Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.
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3
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1, 1, 2, 1, 6, 3, 1, 12, 18, 4, 1, 20, 60, 40, 5, 1, 30, 150, 200, 75, 6, 1, 42, 315, 700, 525, 126, 7, 1, 56, 588, 1960, 2450, 1176, 196, 8, 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums = A001700: (1, 3, 10, 35, 126,...).
Also a(n,k) = binomial[n - 1, k - 1]*binomial[n, k - 1], related to Narayana polynomials (see Sulanke reference). - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 09 2008
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REFERENCES
| Sulanke, R. A. "Counting Lattice Paths by Narayana Polynomials." Electronic J. Combinatorics 7, No. 1, R40, 1-9, 2000. http://www.combinatorics.org/Volume_7/Abstracts/v7i1r40.html.
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FORMULA
| T(n,k)= (k+1)*binomial(n+1,k+1)*binomial(n+1,k)/(n+1), n>=k>=0.
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 14 2010: (Start)
p(x,n)=(1 - x)^(2*n)*Sum[Binomial[k + n - 1, k]*Binomial[n + k, k]*x^k, {k, 0, Infinity}];
p(x,n)=(1 - x)^(2n) HypergeometricPFQ[{n, 1 + n}, {1}, x];
t(n,m)=coefficients(p(x,n)) (End)
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EXAMPLE
| First few rows of the triangle are:
1;
1, 2;
1, 6, 3;
1, 12, 18, 4;
1, 20, 60, 40, 5;
1, 30, 150, 200, 75, 6;
1, 42, 315, 700, 525, 126, 7,
...
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MATHEMATICA
| A[n_, k_]=Binomial[n-1, k-1]*Binomial[n, k-1]; Table[Table[A[n, k], {k, 1, n}], {n, 1, 11}]; Flatten[%] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 09 2008
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 14 2010: (Start)
Clear[p, x, n, m, f, l];
p[x_, n_] = (1 - x)^(2*n)*Sum[Binomial[k + n - 1, k]*Binomial[n + k, k]*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m]];
l[n_] := Length[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x]];
Table[Table[FullSimplify[ExpandAll[f[n, m]]], {m, 1, l[n]}], {n, 1, 10}];
Flatten[%] (End)
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CROSSREFS
| Cf. A127648, A001263, A001700.
Sequence in context: A142977 A120108 A060556 * A034898 A059300 A046803
Adjacent sequences: A132810 A132811 A132812 * A132814 A132815 A132816
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2007
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