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%I #10 Mar 13 2024 13:25:52
%S 1,0,1,0,4,1,0,28,12,1,0,280,160,24,1,0,3640,2520,520,40,1,0,58240,
%T 46480,11880,1280,60,1,0,1106560,987840,295960,40040,2660,84,1,0,
%U 24344320,23826880,8090880,1296960,109200,4928,112,1
%N 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.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
%F T(n, k) = A371076(n, k) / k!.
%F From _Werner Schulte_, Mar 13 2024: (Start)
%F T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
%F 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)
%e Triangle starts:
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 4, 1;
%e [3] 0, 28, 12, 1;
%e [4] 0, 280, 160, 24, 1;
%e [5] 0, 3640, 2520, 520, 40, 1;
%e [6] 0, 58240, 46480, 11880, 1280, 60, 1;
%e [7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;
%p a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
%p # Alternative:
%p BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x),x, 22), x, n), 9);
%p # Recurrence:
%p T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
%p T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
%p for n from 0 to 7 do seq(T(n, k), k = 0..n) od; # _Peter Luschny_, Mar 13 2024
%Y Variant: A035469.
%Y Cf. A264428, A371076, A371077.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Mar 12 2024
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