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A104713
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Triangle T(n,k) = binomial(n,k), read by rows, 3 <= k <=n .
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2
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1, 4, 1, 10, 5, 1, 20, 15, 6, 1, 35, 35, 21, 7, 1, 56, 70, 56, 28, 8, 1, 84, 126, 126, 84, 36, 9, 1, 120, 210, 252, 210, 120, 45, 10, 1, 165, 330, 462, 462, 330, 165, 55, 11, 1, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1, 286, 715, 1287, 1716
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OFFSET
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3,2
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LINKS
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FORMULA
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T(n,k) = A007318(n,k) for n>=3, 3<=k<=n.
The following remarks assume an offset of 0.
Riordan array (1/(1 - x)^4, x/(1 - x)).
O.g.f.: 1/(1 - t)^3*1/(1 - (1 + x)*t) = 1 + (4 + x)*t + (10 + 5*x + x^2)*t^2 + ....
E.g.f.: (1/x*d/dt)^3 (exp(t)*(exp(x*t) - 1 - x*t - (x*t)^2/2!)) = 1 + (4 + x)*t + (10 + 5*x + x^2)*t^2/2! + ....
The infinitesimal generator for this triangle has the sequence [4,5,6,...] on the main subdiagonal and 0's elsewhere. (End)
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EXAMPLE
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First few rows of the triangle are:
1;
4, 1;
10, 5, 1;
20, 15, 6, 1;
35, 35, 21, 7, 1;
56, 70, 56, 28, 8, 1;
...
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MAPLE
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binomial(n, k) ;
end proc;
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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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