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

Author

Peter Luschny, Dec 06 2015

Keywords

• nonn
• tabl

Examples

			[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.
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Peter Luschny, The Bell transform

Crossrefs

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

Programs

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 *)

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

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))