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%I #23 Oct 21 2023 05:35:52
%S 1,0,0,1,0,1,0,1,6,0,1,20,0,1,50,75,0,1,112,525,0,1,238,2450,1575,0,1,
%T 492,9590,18900,0,1,1002,34125,141750,49140,0,1,2024,114675,854700,
%U 900900,0,1,4070,371580,4544925,9909900,2110185
%N Triangle read by rows. T(n, k) = Bell(k)*Sum_{j=0..k}(-1)^(k+j)*binomial(n, n-k+j)*Stirling2(n-k+j, j) for n >= 0 and 0 <= k <= floor(n/2).
%F T(n, k) = (-1)^k*A000110(k)*A137375(n, k) = A000110(k)*A008299(n, k).
%F T(2*n, n) = A081066(n).
%F E.g.f. column k: Bell(k)*(exp(x) - 1 - x)^k / k!, k >= 0.
%F T(n, k) = Bell(k)*Sum_{j=0..k} Sum_{i=0..j} ((-1)^j*(k-j)^(n-i)*binomial(n, i)) / ((k - j)!*(j - i)!).
%e Triangle starts:
%e [0] 1;
%e [1] 0;
%e [2] 0, 1;
%e [3] 0, 1;
%e [4] 0, 1, 6;
%e [5] 0, 1, 20;
%e [6] 0, 1, 50, 75;
%e [7] 0, 1, 112, 525;
%e [8] 0, 1, 238, 2450, 1575;
%e [9] 0, 1, 492, 9590, 18900;
%p A352607 := (n, k) -> combinat:-bell(k)*add((-1)^(k+j)*binomial(n, n-k+j)* Stirling2(n-k+j, j), j = 0..k):
%p seq(seq(A352607(n, k), k = 0..n/2), n = 0..12);
%p # Second program:
%p egf := k -> combinat[bell](k)*(exp(x) - 1 - x)^k/k!:
%p A352607 := (n, k) -> n! * coeff(series(egf(k), x, n+1), x, n):
%p seq(print(seq(A352607(n, k), k = 0..n/2)), n=0..12);
%p # Recurrence:
%p A352607 := proc(n, k) option remember;
%p if k > n/2 then 0 elif k = 0 then k^n else k*A352607(n-1, k) +
%p combinat[bell](k)/combinat[bell](k-1)*(n-1)*A352607(n-2, k-1) fi end:
%p seq(print(seq(A352607(n, k), k=0..n/2)), n=0..12); # _Mélika Tebni_, Mar 24 2022
%t T[n_, k_] := BellB[k]*Sum[(-1)^(k+j)*Binomial[n, n-k+j]*StirlingS2[n-k+j, j], {j, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten (* _Jean-François Alcover_, Oct 21 2023 *)
%Y Cf. A028248 (row sums), A052515 (column 2), A081066, A008299, A000110, A137375.
%K nonn,tabl
%O 0,9
%A _Peter Luschny_ and _Mélika Tebni_, Mar 23 2022
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