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

0,5

Links

Table of n, a(n) for n=0..74.

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

Example

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 ...

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

Crossrefs

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

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

Sequence in context: A144042 A122084 A104559 * A071922 A138028 A009999

Adjacent sequences: A080850 A080851 A080852 * A080854 A080855 A080856

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 23 2003

Status

approved