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A352363 Triangle read by rows. The incomplete Bell transform of the swinging factorials A056040.

%I A352363 #9 Mar 15 2022 08:17:02
%S A352363 1,0,1,0,1,1,0,2,3,1,0,6,11,6,1,0,6,50,35,10,1,0,30,166,225,85,15,1,0,
%T A352363 20,756,1246,735,175,21,1,0,140,2932,7588,5761,1960,322,28,1,0,70,
%U A352363 11556,45296,46116,20181,4536,546,36,1
%N A352363 Triangle read by rows. The incomplete Bell transform of the swinging factorials A056040.
%F A352363 Given a sequence s let s|n denote the initial segment s(0), s(1), ..., s(n).
%F A352363 (T(s))(n, k) = IncompleteBellPolynomial(n, k, s|n), where s(n) = n!/floor(n/2)!^2.
%e A352363 Triangle starts:
%e A352363 [0] 1;
%e A352363 [1] 0,   1;
%e A352363 [2] 0,   1,     1;
%e A352363 [3] 0,   2,     3,     1;
%e A352363 [4] 0,   6,    11,     6,     1;
%e A352363 [5] 0,   6,    50,    35,    10,     1;
%e A352363 [6] 0,  30,   166,   225,    85,    15,    1;
%e A352363 [7] 0,  20,   756,  1246,   735,   175,   21,   1;
%e A352363 [8] 0, 140,  2932,  7588,  5761,  1960,  322,  28, 1;
%e A352363 [9] 0,  70, 11556, 45296, 46116, 20181, 4536, 546, 36, 1;
%p A352363 SwingNumber := n -> n! / iquo(n, 2)!^2:
%p A352363 for n from 0 to 9 do
%p A352363 seq(IncompleteBellB(n, k, seq(SwingNumber(j), j = 0..n)), k = 0..n) od;
%Y A352363 Cf. A056040, A352364 (row sums), A352365 (alternating row sums).
%Y A352363 Cf. A352366, A352369.
%K A352363 nonn,tabl
%O A352363 0,8
%A A352363 _Peter Luschny_, Mar 15 2022