A265023 -id:A265023 - cp's OEIS frontend

A265313 Square array read by ascending antidiagonals, complementary Bell numbers iterated by the Bell transform.

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, 2, -6, 22, -22, -9, 1, 1, -1, 2, -6, 24, -95, 54, -9, 1, 1, -1, 2, -6, 24, -118, 472, -139, 50, 1, 1, -1, 2, -6, 24, -120, 683, -2638, 372, 267, 1, 1, -1, 2, -6, 24

Offset: 0

Peter Luschny, Dec 06 2015

			[ 1,  1, 1,  1,  1,   1,    1,     1,     1, ...] A000012
[ 1, -1, 0,  1,  1,  -2,   -9,    -9,    50, ...] A000587
[ 1, -1, 2, -4,  9, -22,   54,  -139,   372, ...] A265023
[ 1, -1, 2, -6, 22, -95,  472, -2638, 16343, ...]
[ 1, -1, 2, -6, 24, -118, 683, -4533, 33862, ...]
[ 1, -1, 2, -6, 24, -120, 718, -4989, 39405, ...]
[...                                         ...]
[ 1, -1, 2, -6, 24, -120, 720, -5040, 40320, ...] A133942

Peter Luschny, The Bell transform

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

# uses[bell_transform from A264428]
def complementary_bell_number_matrix(ord, len):
    b = [1]*len; L = [b]
    for k in (1..ord-1):
        b = [sum((-1)^n*c for (n, c) in enumerate(bell_transform(n, b))) for n in range(len)]
        L.append(b)
    return matrix(ZZ, L)
print(complementary_bell_number_matrix(6,9))

A265022 Row sums of the Bell transform of the complementary Bell numbers (A264435).

Original entry on oeis.org

1, 1, 0, -2, -1, 12, 20, -113, -430, 1278, 10821, -10234, -317048, -384915, 10352420, 42836466, -340348905, -3180089128, 8045616512, 219303897655, 301713947470, -14401913182942, -84197219028827, 824481606288554, 11426928115546036, -23133559937561187

Offset: 0

Peter Luschny, Dec 01 2015

sign

Peter Luschny, The Bell transform

Cf. A000587 (complementary Bell numbers), A264428, A264435, A265023, A265024.

Table[Sum[BellY[n, k, BellB[Range[n] - 1, -1]], {k, 0, n}], {n, 0, 30}] (* Vladimir Reshetnikov, Nov 09 2016 *)

# uses[bell_transform from A264428]
def A265022_list(len):
    uno = [1]*len
    complementary_bell_numbers = [sum((-1)^n*b for (n, b) in enumerate (bell_transform(n, uno))) for n in range(len)]
    return [sum(bell_transform(n, complementary_bell_numbers)) for n in range(len)]
A265022_list(26)

cp's OEIS Frontend

A265313 Square array read by ascending antidiagonals, complementary Bell numbers iterated by the Bell transform.

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