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

1,5

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.

Links

Table of n, a(n) for n=1..66.

Example

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.

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

Crossrefs

Cf. A001629 (antidiagonal sums), A104249.

Sequence in context: A218618 A271451 A131248 * A110291 A152027 A077308

Adjacent sequences: A116442 A116443 A116444 * A116446 A116447 A116448

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Feb 15 2006

Extensions

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

Status

approved