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A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals.

%I A086270 #57 Sep 07 2024 15:44:33
%S A086270 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 A086270 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 A086270 27,11,1,66,100,117,120,112,96,75,52,30,12,1,78,121,145,153,148,133,111
%N A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals.
%C A086270 The antidiagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.
%C A086270 This is the accumulation array (cf. A144112) of A144257 (which is the weight array of this sequence). - _Clark Kimberling_, Sep 16 2008
%C A086270 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
%D A086270 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 76 at p. 189.
%D A086270 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.
%H A086270 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 A086270 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polygonal_number#Table_of_values">Polygonal number: Table of values</a>.
%F A086270 T(n, k) = n*binomial(k, 2) + k = A057145(n+2,k).
%F A086270 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 A086270 From _Stefano Spezia_, Sep 02 2022: (Start)
%F A086270 G.f.: x*y*(1 - x + x*y)/((1 - x)^2*(1 - y)^3).
%F A086270 G.f. of k-th column: k*(1 + k - 2*x)*x/(2*(1 - x)^2). (End)
%e A086270 First 6 rows:
%e A086270 =========================================
%e A086270 n\k|  1   2    3    4    5    6     7
%e A086270 ---|-------------------------------------
%e A086270 1  |  1   3    6   10   15   21    28 ... (A000217, triangular numbers)
%e A086270 2  |  1   4    9   16   25   36    49 ... (A000290, squares)
%e A086270 3  |  1   5   12   22   35   51    70 ... (A000326, pentagonal numbers)
%e A086270 4  |  1   6   15   28   45   66    91 ... (A000384, hexagonal numbers)
%e A086270 5  |  1   7   18   34   55   81   112 ... (A000566, heptagonal numbers)
%e A086270 6  |  1   8   21   40   65   96   133 ... (A000567, octagonal numbers)
%e A086270 ...
%e A086270 The array formed by the complements: A183225.
%t A086270 t[n_, k_] := n*Binomial[k, 2] + k; Table[ t[k, n - k + 1], {n, 12}, {k, n}] // Flatten
%o A086270 (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 A086270 Cf. A006522, A057145, A086271, A086272, A086273, A139601, A183225.
%Y A086270 Cf. A000217, A000290, A000326, A000384, A000566, A000567.
%Y A086270 Cf. A114112, A144257 .
%K A086270 nonn,easy,tabl
%O A086270 1,2
%A A086270 _Clark Kimberling_, Jul 14 2003
%E A086270 Extended by _Clark Kimberling_, Jan 01 2011