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A367564
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Triangular array read by rows: T(n, k) = binomial(n, k) * A001333(n - k).
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0
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1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 17, 28, 18, 4, 1, 41, 85, 70, 30, 5, 1, 99, 246, 255, 140, 45, 6, 1, 239, 693, 861, 595, 245, 63, 7, 1, 577, 1912, 2772, 2296, 1190, 392, 84, 8, 1, 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1, 3363, 13930, 25965, 28680, 20790, 10332, 3570, 840, 135, 10, 1
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OFFSET
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0,4
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LINKS
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FORMULA
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The row polynomials p(n, x) = Sum_{k=0..n} T(n, k) * x^k satisfy:
a) p'(n, x) = n * p(n-1, x) where p' is the first derivative of p;
b) p(0, x) = 1, p(1, x) = 1 + x and p(n, x) = (2+2*x) * p(n-1, x) + (1-2*x-x^2) * p(n-2, x) for n > 1.
T(n, 0) = A001333(n) for n >= 0 and T(n, k) = T(n-1, k-1) * n / k for 0 < k <= n.
G.f.: (1 - (1+x) * t) / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)
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EXAMPLE
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Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 3, 2, 1;
[3] 7, 9, 3, 1;
[4] 17, 28, 18, 4, 1;
[5] 41, 85, 70, 30, 5, 1;
[6] 99, 246, 255, 140, 45, 6, 1;
[7] 239, 693, 861, 595, 245, 63, 7, 1;
[8] 577, 1912, 2772, 2296, 1190, 392, 84, 8, 1;
[9] 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1;
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MAPLE
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P := proc(n) option remember; ifelse(n <= 1, 1, 2*P(n - 1) + P(n - 2)) end:
T := (n, k) -> P(n - k) * binomial(n, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
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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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