A116445 Array read by antidiagonals: the binomial transform of the sequence (1,2,..n,0,0,0..) in row n.

1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 8, 7, 1, 1, 3, 8, 16, 9, 1, 1, 3, 8, 20, 27, 11, 1, 1, 3, 8, 20, 43, 41, 13, 1, 1, 3, 8, 20, 48, 81, 58, 15, 1, 1, 3, 8, 20, 48, 106, 138, 78, 17, 1, 1, 3, 8, 20, 48, 112, 213, 218, 101, 19, 1

Offset: 1

Author

Gary W. Adamson, Feb 15 2006

Keywords

• nonn
• tabl
• easy

Comments

Create an array by rows: (binomial transforms of 1,0,0,0,...; 1,2,0,0,0,...; 1,2,3,0,0,0,...; etc.). Antidiagonals of the array become rows of the triangle.

Examples

			First few rows of the array:
  1, 1, 1,  1,  1,   1,   1,   1,   1, 1, ...
  1, 3, 5,  7,  9,  11,  13,  15,  17, ...
  1, 3, 8, 16, 27,  41,  58,  78, 101, ...  A104249
  1, 3, 8, 20, 43,  81, 138, 218, ...       A139488
  1, 3, 8, 20, 48, 106, 213, ...
  1, 3, 8, 20, 48, 112, 249, ...
  ...
Diagonals converge to A001792, binomial transform of (1,2,3,...); and the first few rows of the triangle created by reading upwards antidiagonals are:
  1
  1, 1;
  1, 3, 1;
  1, 3, 5,  1;
  1, 3, 8,  7,  1;
  1, 3, 8, 16,  9,  1;
  1, 3, 8, 20, 27, 22, 1;
  ...
a(4), a(5), a(6) = 1, 3, 1 = antidiagonals of the array becoming row three of the triangle.
		

Crossrefs

Cf. A001629 (antidiagonal sums), A104249.

Programs

Maple

A116445 := proc(n,k)
    local a,i ;
    a := 0 ;
    for i from 0 to n do
        a := a+binomial(k,i)*(i+1) ;
    end do:
    a ;
end proc:
seq(seq(A116445(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Aug 17 2022

Extensions

Detailed NAME by R. J. Mathar, Aug 17 2022