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A352366 Triangle read by rows. The incomplete Bell transform of the Catalan numbers.

%I A352366 #11 Mar 15 2022 08:17:31
%S A352366 1,0,1,0,1,1,0,2,3,1,0,5,11,6,1,0,14,45,35,10,1,0,42,199,210,85,15,1,
%T A352366 0,132,938,1309,700,175,21,1,0,429,4675,8498,5789,1890,322,28,1,0,
%U A352366 1430,24489,57455,48762,19929,4410,546,36,1
%N A352366 Triangle read by rows. The incomplete Bell transform of the Catalan numbers.
%F A352366 Given a sequence s let s|n denote the initial segment s(0), s(1), ..., s(n).
%F A352366 (T(s))(n, k) = IncompleteBellPolynomial(n, k, s|n) where s(n) = CatalanNumber(n).
%e A352366 Triangle start:
%e A352366 [0] 1;
%e A352366 [1] 0,    1;
%e A352366 [2] 0,    1,     1;
%e A352366 [3] 0,    2,     3,     1;
%e A352366 [4] 0,    5,    11,     6,     1;
%e A352366 [5] 0,   14,    45,    35,    10,     1;
%e A352366 [6] 0,   42,   199,   210,    85,    15,    1;
%e A352366 [7] 0,  132,   938,  1309,   700,   175,   21,   1;
%e A352366 [8] 0,  429,  4675,  8498,  5789,  1890,  322,  28,  1;
%e A352366 [9] 0, 1430, 24489, 57455, 48762, 19929, 4410, 546, 36, 1;
%p A352366 CatalanNumber := n -> binomial(2*n, n)/(n + 1):
%p A352366 for n from 0 to 9 do
%p A352366 seq(IncompleteBellB(n, k, seq(CatalanNumber(j), j=0 .. n)), k = 0..n) od;
%Y A352366 Cf. A000108, A352367 (row sums), A352368 (alternating row sums).
%Y A352366 Cf. A352363, A352369.
%K A352366 nonn,tabl
%O A352366 0,8
%A A352366 _Peter Luschny_, Mar 15 2022