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A371080 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.

%I A371080 #19 Apr 19 2025 10:03:34
%S A371080 1,0,1,0,4,1,0,28,12,1,0,280,160,24,1,0,3640,2520,520,40,1,0,58240,
%T A371080 46480,11880,1280,60,1,0,1106560,987840,295960,40040,2660,84,1,0,
%U A371080 24344320,23826880,8090880,1296960,109200,4928,112,1
%N A371080 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 A371080 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F A371080 T(n, k) = BellMatrix([x^n] hypergeom2F0([1, 1/3], [], 3*x) / x).
%F A371080 T(n, k) = A371076(n, k) / k!.
%F A371080 From _Werner Schulte_, Mar 13 2024: (Start)
%F A371080 T(n, k) = (Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * Product_{j=0..n-1} (3*j + i)) / (k!).
%F A371080 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)
%F A371080 From _Seiichi Manyama_, Apr 19 2025: (Start)
%F A371080 T(n,k) = Sum_{j=k..n} 3^(n-j) * |Stirling1(n,j)| * Stirling2(j,k).
%F A371080 E.g.f. of column k (with leading zeros): (1/(1 - 3*x)^(1/3) - 1)^k / k!. (End)
%e A371080 Triangle starts:
%e A371080 [0] 1;
%e A371080 [1] 0,       1;
%e A371080 [2] 0,       4,      1;
%e A371080 [3] 0,      28,     12,      1;
%e A371080 [4] 0,     280,    160,     24,     1;
%e A371080 [5] 0,    3640,   2520,    520,    40,    1;
%e A371080 [6] 0,   58240,  46480,  11880,  1280,   60,  1;
%e A371080 [7] 0, 1106560, 987840, 295960, 40040, 2660, 84, 1;
%p A371080 a := n -> mul(select(k -> k mod 3 = 1, [seq(1..3*(n + 1))])): BellMatrix(a, 9);
%p A371080 # Alternative:
%p A371080 BellMatrix(n -> coeff(series((1/x)*hypergeom([1, 1/3], [], 3*x),x, 22), x, n), 9);
%p A371080 # Recurrence:
%p A371080 T := proc(n, k) option remember; if k = n then 1 elif k = 0 then 0 else
%p A371080 T(n - 1, k - 1) + (3*(n - 1) + k) * T(n - 1, k) fi end:
%p A371080 for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # _Peter Luschny_, Mar 13 2024
%o A371080 (PARI) T(n, k) = sum(j=k, n, 3^(n-j)*abs(stirling(n, j, 1))*stirling(j, k, 2)); \\ _Seiichi Manyama_, Apr 19 2025
%Y A371080 Variant: A035469.
%Y A371080 Cf. A264428, A371076, A371077.
%Y A371080 Cf. A007559, A144346.
%Y A371080 Cf. A035342, A049029, A271703.
%K A371080 nonn,tabl
%O A371080 0,5
%A A371080 _Peter Luschny_, Mar 12 2024