A123081 Infinite square array read by antidiagonals: T(n,k) = Bell(n+k) = A000110(n+k).

1, 1, 1, 2, 2, 2, 5, 5, 5, 5, 15, 15, 15, 15, 15, 52, 52, 52, 52, 52, 52, 203, 203, 203, 203, 203, 203, 203, 877, 877, 877, 877, 877, 877, 877, 877, 4140, 4140, 4140, 4140, 4140, 4140, 4140, 4140, 4140, 21147, 21147, 21147, 21147, 21147, 21147, 21147, 21147, 21147, 21147, 115975, 115975, 115975, 115975

Offset

0,4

Comments

Alternatively, triangle read by rows in which row n (n >= 0) contains A000110(n) repeated n+1 times.

Row sums = A052887: 1, 2, 6, 20, 75, 312, ... A127568 = Q * M n-th row is composed of n+1 terms of A000110(n).

Links

G. C. Greubel, Antidiagonal rows n = 0..50, flattened

W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16. [From N. J. A. Sloane, Feb 07 2009]

Formula

M * Q, as infinite lower triangular matrices; M = the Bell sequence, A000110 in the main diagonal and the rest zeros. Q = (1; 1, 1; 1, 1, 1; ...)

Example

Square array begins:
    1,    1,     2,      5,     15,      52,      203,       877, ...;
    1,    2,     5,     15,     52,     203,      877,      4140, ...;
    2,    5,    15,     52,    203,     877,     4140,     21147, ...;
    5,   15,    52,    203,    877,    4140,    21147,    115975, ...;
   15,   52,   203,    877,   4140,   21147,   115975,    678570, ...;
   52,  203,   877,   4140,  21147,  115975,   678570,   4213597, ...;
  203,  877,  4140,  21147, 115975,  678570,  4213597,  27644437, ...;
  877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, ...;
First few rows of the triangle:
    1;
    1,   1;
    2,   2,   2;
    5,   5,   5,   5;
   15,  15,  15,  15,  15;
   52,  52,  52,  52,  52,  52;
  203, 203, 203, 203, 203, 203, 203;

Mathematica

Table[BellB[n], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 21 2021 *)

Prog

(PARI) B(n)=sum(k=0, n, stirling(n, k, 2));
for(n=0, 20, for(k=0, n, print1(B(n), ", "))); \\ Joerg Arndt, Apr 21 2014
(Magma) [Bell(n): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2021
(Sage) flatten([[bell_number(n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 21 2021

Crossrefs

Cf. A000110, A052887, A127568.

Sequence in context: A105960 A081290 A168256 * A020917 A308772 A332966

Adjacent sequences: A123078 A123079 A123080 * A123082 A123083 A123084

Keyword

nonn,easy,tabl

Author

Gary W. Adamson, Jan 19 2007

Extensions

Edited by N. J. A. Sloane, Feb 07 2009

Added more terms, Joerg Arndt, Apr 21 2014

Status

approved