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A371080
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Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.
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1
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1, 0, 1, 0, 4, 1, 0, 28, 12, 1, 0, 280, 160, 24, 1, 0, 3640, 2520, 520, 40, 1, 0, 58240, 46480, 11880, 1280, 60, 1, 0, 1106560, 987840, 295960, 40040, 2660, 84, 1, 0, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
T(n, k) = T(n-1, k-1) + (3*(n - 1) + k) * T(n-1, k) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0. (End)
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EXAMPLE
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Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 4, 1;
[3] 0, 28, 12, 1;
[4] 0, 280, 160, 24, 1;
[5] 0, 3640, 2520, 520, 40, 1;
[6] 0, 58240, 46480, 11880, 1280, 60, 1;
[7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;
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MAPLE
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a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
# Alternative:
BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x), x, 22), x, n), 9);
# Recurrence:
T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Mar 13 2024
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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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