A086271 Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.

1, 1, 3, 1, 4, 6, 1, 5, 9, 10, 1, 6, 12, 16, 15, 1, 7, 15, 22, 25, 21, 1, 8, 18, 28, 35, 36, 28, 1, 9, 21, 34, 45, 51, 49, 36, 1, 10, 24, 40, 55, 66, 70, 64, 45, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 1, 12, 30, 52, 75, 96, 112, 120, 117, 100, 66, 1, 13, 33, 58, 85, 111, 133, 148, 153, 145, 121, 78

Offset: 1

Author

Clark Kimberling, Jul 14 2003

Keywords

• nonn
• easy
• tabl

Comments

The transpose of the array in A086270; diagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.

Examples

			Columns 1,2,3 are the triangular, square and pentagonal numbers.
Northwest corner:
       k=1 k=2 k=3 k=4 k=5
  n=1:   1   1   1   1   1 ...
  n=2:   3   4   5   6   7 ...
  n=3:   6   9  12  15  18 ...
  n=4:  10  16  22  28  34 ...
  n=5:  15  25  35  45  55 ...
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 diagonals, flattened
• Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Crossrefs

Cf. A006522, A086270, A086272, A086273.

Main diagonal gives A006000(n-1).

Programs

Mathematica

T[n_, k_] := PolygonalNumber[k+2, n]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 04 2016 *)

Formula

T(n, k) = k*C(n,2) + n.

From Stefano Spezia, Sep 02 2022: (Start)

G.f.: x*y*(1 - y + x*y)/((1 - x)^3*(1 - y)^2).

G.f. of n-th row: n*(1 + n - 2*y)*y/(2*(1 - y)^2). (End)