A080853 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

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