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A362585 Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).

%I A362585 #11 May 10 2023 11:50:26
%S A362585 1,1,1,3,6,3,13,39,39,13,75,300,450,300,75,541,2705,5410,5410,2705,
%T A362585 541,4683,28098,70245,93660,70245,28098,4683,47293,331051,993153,
%U A362585 1655255,1655255,993153,331051,47293,545835,4366680,15283380,30566760,38208450,30566760,15283380,4366680,545835
%N A362585 Triangle read by rows, T(n, k) = A000670(n) * binomial(n, k).
%e A362585 [0]    1;
%e A362585 [1]    1,     1;
%e A362585 [2]    3,     6,     3;
%e A362585 [3]   13,    39,    39,    13;
%e A362585 [4]   75,   300,   450,   300,    75;
%e A362585 [5]  541,  2705,  5410,  5410,  2705,   541;
%e A362585 [6] 4683, 28098, 70245, 93660, 70245, 28098, 4683;
%o A362585 (SageMath)
%o A362585 def TransOrdPart(m, n) -> list[int]:
%o A362585     @cached_function
%o A362585     def P(m: int, n: int):
%o A362585         R = PolynomialRing(ZZ, "x")
%o A362585         if n == 0: return R(1)
%o A362585         return R(sum(binomial(m * n, m * k) * P(m, n - k) * x
%o A362585                  for k in range(1, n + 1)))
%o A362585     T = P(m, n)
%o A362585     def C(k) -> int:
%o A362585         return sum(T[j] * binomial(n, k) for j in range(n + 1))
%o A362585     return [C(k) for k in range(n+1)]
%o A362585 def A362585(n) -> list[int]: return TransOrdPart(1, n)
%o A362585 for n in range(6): print(A362585(n))
%Y A362585 Family of triangles: A055372 (m=0, Pascal), this sequence (m=1, Fubini), A362586 (m=2, Joffe), A362849 (m=3, A278073).
%Y A362585 Cf. A000670 (column 0 and main diagonal), A216794 (row sums).
%K A362585 nonn,tabl
%O A362585 0,4
%A A362585 _Peter Luschny_, Apr 26 2023