A080853 Square array of generalized polygonal numbers, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 9, 7, 1, 1, 5, 16, 19, 11, 1, 1, 6, 25, 37, 33, 16, 1, 1, 7, 36, 61, 67, 51, 22, 1, 1, 8, 49, 91, 113, 106, 73, 29, 1, 1, 9, 64, 127, 171, 181, 154, 99, 37, 1, 1, 10, 81, 169, 241, 276, 265, 211, 129, 46, 1, 1, 11, 100, 217, 323, 391, 406, 365, 277

Offset: 0

Author

Paul Barry, Feb 23 2003

Keywords

• easy
• nonn
• tabl

Examples

			Rows begin with n>=0, k>=0
1 1 1 1 1 ...
1 2 4 7 11 ...
1 3 9 19 33 ...
1 4 16 37 67 ...
1 5 25 61 113 ...
		

Crossrefs

Rows include A000124, A058331, A080855, A080856, A080857, A080919.

Columns include A000290, A003215, A003215, A080859, A080860, A080861.

Programs

Maple

A080853 := proc(n,k)
    binomial(k,0)+n*binomial(k,1)+n^2*binomial(k,2) ;
end proc:
seq( seq(A080853(d-k,k),k=0..d),d=0..12) ; # R. J. Mathar, Oct 01 2021

Formula

T(n, k)=C(k, 0)+C(k, 1)n+C(k, 2)n^2=(n^2*k^2-(n^2-2n)*k+2)/2 =(k(k-1)*n^2+2k*n+2)/2

Row n has g.f. (1+(n-2)x+(n^2-n+1)x^2)/(1-x)^3.

Column k has g.f. (C(k-1, 0)+(C(k+1, 2)-2)*x+C(k-1, 2)*x^2)/(1-x)^3.

Diagonals are given by (n^4+(2k-1)*n^3+((k-1)^2+1)*n^2+(1-(k-1)^2)*n+2)/2.

Antidiagonal sums are 1, 2, 4, 9, 22, 53, 119,... = (d+1)*(2*d^4-7*d^3+27*d^2-22*d+120)/120 = sum_{k=0..d} T(d-k,k), first differences in A116701, d>=0. - R. J. Mathar, Oct 01 2021