login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A265312 Square array read by ascending antidiagonals, Bell numbers iterated by the Bell transform. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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 A273894 A308035

Adjacent sequences:  A265309 A265310 A265311 * A265313 A265314 A265315

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Dec 06 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 12 09:35 EDT 2020. Contains 335657 sequences. (Running on oeis4.)