A265312 Square array read by ascending antidiagonals, Bell numbers iterated by the Bell transform.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 1, 1, 1, 2, 6, 15, 1, 1, 1, 2, 6, 23, 52, 1, 1, 1, 2, 6, 24, 106, 203, 1, 1, 1, 2, 6, 24, 119, 568, 877, 1, 1, 1, 2, 6, 24, 120, 700, 3459, 4140, 1, 1, 1, 2, 6, 24, 120, 719, 4748, 23544, 21147, 1, 1, 1, 2, 6, 24, 120, 720, 5013, 36403, 176850, 115975, 1

Offset

0,9

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Peter Luschny, The Bell transform

Example

[1, 1, 1, 1,  1,   1,   1,    1,     1, ...] A000012
[1, 1, 2, 5, 15,  52, 203,  877,  4140, ...] A000110
[1, 1, 2, 6, 23, 106, 568, 3459, 23544, ...] A187761
[1, 1, 2, 6, 24, 119, 700, 4748, 36403, ...] A264432
[1, 1, 2, 6, 24, 120, 719, 5013, 39812, ...]
[1, 1, 2, 6, 24, 120, 720, 5039, 40285, ...]
[...                                    ...]
[1, 1, 2, 6, 24, 120, 720, 5040, 40320, ...] A000142 = main diagonal.

Maple

A:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
      binomial(n-1, j-1)*A(j-1, h-1)*A(n-j, h), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 21 2017

Mathematica

A[n_, h_]:=A[n, h]=If[Min[n, h]==0, 1, Sum[Binomial[n - 1, j - 1] A[j - 1, h - 1] A[n - j, h] , {j, n}]]; Table[A[n, d - n], {d, 0, 12}, {n, 0, d}]//Flatten (* Indranil Ghosh, Aug 21 2017, after maple code *)

Prog

(Sage) # uses[bell_transform from A264428]
def bell_number_matrix(ord, len):
    b = [1]*len; L = [b]
    for k in (1..ord-1):
        b = [sum(bell_transform(n, b)) for n in range(len)]
        L.append(b)
    return matrix(ZZ, L)
print(bell_number_matrix(6, 9))
(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def A(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*A(j - 1, h - 1)*A(n - j, h) for j in range(1, n + 1)])
for d in range(13): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Aug 21 2017, after Maple code

Crossrefs

Cf. A000012, A000110, A000142, A187761, A264428, A264432, A265313.

Sequence in context: A215894 A061545 A287641 * A241531 A362277 A367955

Adjacent sequences: A265309 A265310 A265311 * A265313 A265314 A265315

Keyword

nonn,tabl

Author

Peter Luschny, Dec 06 2015

Status

approved