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

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, 36, 49, 51, 45, 34, 21, 9, 1, 45, 64, 70, 66, 55, 40, 24, 10, 1, 55, 81, 92, 91, 81, 65, 46, 27, 11, 1, 66, 100, 117, 120, 112, 96, 75, 52, 30, 12, 1, 78, 121, 145, 153, 148, 133, 111

Offset

1,2

Comments

The antidiagonal sums 1, 4, 11, 25, 50, ... are the numbers A006522(n) for n >= 3.

This is the accumulation array (cf. A144112) of A144257 (which is the weight array of this sequence). - Clark Kimberling, Sep 16 2008

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

References

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 76 at p. 189.

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 123.

Links

Table of n, a(n) for n=1..73.

Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Wikipedia, Polygonal number: Table of values.

Formula

T(n, k) = n*binomial(k, 2) + k = A057145(n+2,k).

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

From Stefano Spezia, Sep 02 2022: (Start)

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

G.f. of k-th column: k*(1 + k - 2*x)*x/(2*(1 - x)^2). (End)

Example

First 6 rows:
=========================================
n\k|  1   2    3    4    5    6     7
---|-------------------------------------
1  |  1   3    6   10   15   21    28 ... (A000217, triangular numbers)
2  |  1   4    9   16   25   36    49 ... (A000290, squares)
3  |  1   5   12   22   35   51    70 ... (A000326, pentagonal numbers)
4  |  1   6   15   28   45   66    91 ... (A000384, hexagonal numbers)
5  |  1   7   18   34   55   81   112 ... (A000566, heptagonal numbers)
6  |  1   8   21   40   65   96   133 ... (A000567, octagonal numbers)
...
The array formed by the complements: A183225.

Mathematica

t[n_, k_] := n*Binomial[k, 2] + k; Table[ t[k, n - k + 1], {n, 12}, {k, n}] // Flatten

Prog

(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

Crossrefs

Cf. A006522, A057145, A086271, A086272, A086273, A139601, A183225.

Cf. A000217, A000290, A000326, A000384, A000566, A000567.

Cf. A114112, A144257 .

Sequence in context: A133110 A286158 A185915 * A325000 A104712 A122177

Adjacent sequences: A086267 A086268 A086269 * A086271 A086272 A086273

Keyword

nonn,easy,tabl

Author

Clark Kimberling, Jul 14 2003

Extensions

Extended by Clark Kimberling, Jan 01 2011

Status

approved