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Rectangular array T(n,k) of polygonal numbers, by descending antidiagonals.
9

%I #47 Aug 31 2023 16:13:53

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

%C The transpose of the array in A086270; diagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.

%H G. C. Greubel, <a href="/A086271/b086271.txt">Table of n, a(n) for the first 50 diagonals, flattened</a>

%H 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 T(n, k) = k*C(n,2) + n.

%F From _Stefano Spezia_, Sep 02 2022: (Start)

%F G.f.: x*y*(1 - y + x*y)/((1 - x)^3*(1 - y)^2).

%F G.f. of n-th row: n*(1 + n - 2*y)*y/(2*(1 - y)^2). (End)

%e Columns 1,2,3 are the triangular, square and pentagonal numbers.

%e Northwest corner:

%e k=1 k=2 k=3 k=4 k=5

%e n=1: 1 1 1 1 1 ...

%e n=2: 3 4 5 6 7 ...

%e n=3: 6 9 12 15 18 ...

%e n=4: 10 16 22 28 34 ...

%e n=5: 15 25 35 45 55 ...

%e ...

%t 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 Cf. A006522, A086270, A086272, A086273.

%Y Main diagonal gives A006000(n-1).

%K nonn,easy,tabl

%O 1,3

%A _Clark Kimberling_, Jul 14 2003