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

Author

Gary W. Adamson, Jan 19 2007

Keywords

• nonn
• easy
• tabl

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).

Examples

			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;
		

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]

Crossrefs

Cf. A000110, A052887, A127568.

Programs

Magma

[Bell(n): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 21 2021

Mathematica

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

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

Sage

flatten([[bell_number(n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 21 2021

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; ...)

Extensions

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

Added more terms, Joerg Arndt, Apr 21 2014