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

%I A086271 #47 Aug 31 2023 16:13:53
%S A086271 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,
%T A086271 1,9,21,34,45,51,49,36,1,10,24,40,55,66,70,64,45,1,11,27,46,65,81,91,
%U A086271 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
%N A086271 Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.
%C A086271 The transpose of the array in A086270; diagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.
%H A086271 G. C. Greubel, <a href="/A086271/b086271.txt">Table of n, a(n) for the first 50 diagonals, flattened</a>
%H A086271 Clark Kimberling and John E. Brown, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%F A086271 T(n, k) = k*C(n,2) + n.
%F A086271 From _Stefano Spezia_, Sep 02 2022: (Start)
%F A086271 G.f.: x*y*(1 - y + x*y)/((1 - x)^3*(1 - y)^2).
%F A086271 G.f. of n-th row: n*(1 + n - 2*y)*y/(2*(1 - y)^2). (End)
%e A086271 Columns 1,2,3 are the triangular, square and pentagonal numbers.
%e A086271 Northwest corner:
%e A086271        k=1 k=2 k=3 k=4 k=5
%e A086271   n=1:   1   1   1   1   1 ...
%e A086271   n=2:   3   4   5   6   7 ...
%e A086271   n=3:   6   9  12  15  18 ...
%e A086271   n=4:  10  16  22  28  34 ...
%e A086271   n=5:  15  25  35  45  55 ...
%e A086271   ...
%t A086271 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 *)
%Y A086271 Cf. A006522, A086270, A086272, A086273.
%Y A086271 Main diagonal gives A006000(n-1).
%K A086271 nonn,easy,tabl
%O A086271 1,3
%A A086271 _Clark Kimberling_, Jul 14 2003