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%I #7 Oct 01 2021 09:07:09
%S 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,
%T 36,61,67,51,22,1,1,8,49,91,113,106,73,29,1,1,9,64,127,171,181,154,99,
%U 37,1,1,10,81,169,241,276,265,211,129,46,1,1,11,100,217,323,391,406,365,277
%N Square array of generalized polygonal numbers, read by antidiagonals.
%F 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
%F Row n has g.f. (1+(n-2)x+(n^2-n+1)x^2)/(1-x)^3.
%F Column k has g.f. (C(k-1, 0)+(C(k+1, 2)-2)*x+C(k-1, 2)*x^2)/(1-x)^3.
%F Diagonals are given by (n^4+(2k-1)*n^3+((k-1)^2+1)*n^2+(1-(k-1)^2)*n+2)/2.
%F 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
%e Rows begin with n>=0, k>=0
%e 1 1 1 1 1 ...
%e 1 2 4 7 11 ...
%e 1 3 9 19 33 ...
%e 1 4 16 37 67 ...
%e 1 5 25 61 113 ...
%p A080853 := proc(n,k)
%p binomial(k,0)+n*binomial(k,1)+n^2*binomial(k,2) ;
%p end proc:
%p seq( seq(A080853(d-k,k),k=0..d),d=0..12) ; # _R. J. Mathar_, Oct 01 2021
%Y Rows include A000124, A058331, A080855, A080856, A080857, A080919.
%Y Columns include A000290, A003215, A003215, A080859, A080860, A080861.
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Feb 23 2003
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