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%I #51 Sep 04 2022 09:47:15
%S 1,3,1,6,4,1,10,9,5,1,15,16,12,6,1,21,25,22,15,7,1,28,36,35,28,18,8,1,
%T 36,49,51,45,34,21,9,1,45,64,70,66,55,40,24,10,1,55,81,92,91,81,65,46,
%U 27,11,1,66,100,117,120,112,96,75,52,30,12,1,78,121,145,153,148,133,111
%N Rectangular array T(k,n) of polygonal numbers, by antidiagonals.
%C The antidiagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.
%C This is the accumulation array (cf. A144112) of A144257 (which is the weight array of this sequence). - _Clark Kimberling_, Sep 16 2008
%C By rows, the sequence beginning (1, N, ...) is the binomial transform of (1, (N-1), (N-2), 0, 0, 0, ...); and is the second partial sum of (1, (N-2), (N-2), (N-2), ...). Example: The sequence (1, 4, 9, 16, 25, ...) is the binomial transform of (1, 3, 2, 0, 0, 0, ...) and the second partial sum of (1, 2, 2, 2, ...). - _Gary W. Adamson_, Aug 23 2015
%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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number#Table_of_values">Polygonal number: Table of values</a>.
%F T(n, k) = n*binomial(k, 2) + k = A057145(n+2,k).
%F 2*T(n, k) = T(n+r, k) + T(n-r, k), where r = 0, 1, 2, 3, ..., n-1 (see table in Example field). - _Bruno Berselli_, Dec 19 2014
%F From _Stefano Spezia_, Sep 02 2022: (Start)
%F G.f.: x*y*(1 - x + x*y)/((1 - x)^2*(1 - y)^3).
%F G.f. of k-th column: k*(1 + k - 2*x)*x/(2*(1 - x)^2). (End)
%e First 6 rows:
%e =========================================
%e n\k| 1 2 3 4 5 6 7
%e ---|-------------------------------------
%e 1 | 1 3 6 10 15 21 28 ... (A000217, triangular numbers)
%e 2 | 1 4 9 16 25 36 49 ... (A000290, squares)
%e 3 | 1 5 12 22 35 51 70 ... (A000326, pentagonal numbers)
%e 4 | 1 6 15 28 45 66 91 ... (A000384, hexagonal numbers)
%e 5 | 1 7 18 34 55 81 112 ... (A000566, heptagonal numbers)
%e 6 | 1 8 21 40 65 96 133 ... (A000567, octagonal numbers)
%e ...
%e The array formed by the complements: A183225.
%t t[n_, k_] := n*Binomial[k, 2] + k; Table[ t[k, n - k + 1], {n, 12}, {k, n}] // Flatten
%o (Magma) T:=func<h,i | h*Binomial(i,2)+i>; [T(k,n-k+1): k in [1..n], n in [1..12]]; // _Bruno Berselli_, Dec 19 2014
%Y Cf. A006522, A057145, A086271, A086272, A086273, A139601, A183225.
%Y Cf. A000217, A000290, A000326, A000384, A000566, A000567.
%Y Cf. A114112, A144257 .
%K nonn,easy,tabl
%O 1,2
%A _Clark Kimberling_, Jul 14 2003
%E Extended by _Clark Kimberling_, Jan 01 2011
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