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A086270 Rectangular array T(k,n) of polygonal numbers, by antidiagonals. 18
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 (list; table; graph; refs; listen; history; text; internal format)
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
LINKS
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
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
Sequence in context: A133110 A286158 A185915 * A325000 A104712 A122177
KEYWORD
nonn,easy,tabl
AUTHOR
Clark Kimberling, Jul 14 2003
EXTENSIONS
Extended by Clark Kimberling, Jan 01 2011
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)