A265313 - OEIS

%I #20 Sep 22 2025 16:01:23

%S 1,1,1,1,-1,1,1,-1,0,1,1,-1,2,1,1,1,-1,2,-4,1,1,1,-1,2,-6,9,-2,1,1,-1,

%T 2,-6,22,-22,-9,1,1,-1,2,-6,24,-95,54,-9,1,1,-1,2,-6,24,-118,472,-139,

%U 50,1,1,-1,2,-6,24,-120,683,-2638,372,267,1,1,-1,2,-6,24

%N Square array read by ascending antidiagonals, complementary Bell numbers iterated by the Bell transform.

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>

%e [ 1,  1, 1,  1,  1,   1,    1,     1,     1, ...] A000012

%e [ 1, -1, 0,  1,  1,  -2,   -9,    -9,    50, ...] A000587

%e [ 1, -1, 2, -4,  9, -22,   54,  -139,   372, ...] A265023

%e [ 1, -1, 2, -6, 22, -95,  472, -2638, 16343, ...]

%e [ 1, -1, 2, -6, 24, -118, 683, -4533, 33862, ...]

%e [ 1, -1, 2, -6, 24, -120, 718, -4989, 39405, ...]

%e [...                                         ...]

%e [ 1, -1, 2, -6, 24, -120, 720, -5040, 40320, ...] A133942

%o (SageMath) # uses[bell_transform from A264428]

%o def complementary_bell_number_matrix(ord, len):

%o     b = [1]*len; L = [b]

%o     for k in (1..ord-1):

%o         b = [sum((-1)^n*c for (n, c) in enumerate(bell_transform(n, b))) for n in range(len)]

%o         L.append(b)

%o     return matrix(ZZ, L)

%o print(complementary_bell_number_matrix(6,9))

%Y Cf. A000012, A000587, A133942, A264428, A265023, A265312.

%K sign,tabl

%O 0,13

%A _Peter Luschny_, Dec 06 2015