A062026 -id:A062026 - cp's OEIS frontend

A004255 n(n+1)(n^2 -3n + 6)/8.

1, 3, 9, 25, 60, 126, 238, 414, 675, 1045, 1551, 2223, 3094, 4200, 5580, 7276, 9333, 11799, 14725, 18165, 22176, 26818, 32154, 38250, 45175, 53001, 61803, 71659, 82650, 94860, 108376, 123288, 139689, 157675, 177345, 198801, 222148, 247494, 274950

Offset: 1

Dennis S. Kluk (mathemagician(AT)ameritech.net)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A060354. Equals (1/2) A062026.

[n*(n+1)*(n^2-3*n+6)/8: n in [1..50]]; // Vincenzo Librandi, Jun 07 2013

lst={};Do[AppendTo[lst, n*(n+1)*(n^2-3*n+6)/8], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
Table[n(n+1)(n^2-3n+6)/8,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,9,25,60},40] (* Harvey P. Dale, May 27 2023 *)

a(n)=n*(n+1)*(n^2-3*n+6)/8 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: -x*(1-2*x+4*x^2) / (x-1)^5. - Simon Plouffe in his 1992 dissertation.

A109876 Triangle read by rows: a(n, n) = n! and for 1 <= k < n, a(n, k) = Sum_{i=0..n-1} Product_{j=i+1..i+k} f(j, n), where for x <= y, f(x, y) = x and for x > y, f(x, y) = x-y.

Original entry on oeis.org

1, 3, 2, 6, 11, 6, 10, 24, 50, 24, 15, 45, 120, 274, 120, 21, 76, 252, 720, 1764, 720, 28, 119, 476, 1680, 5040, 13068, 5040, 36, 176, 828, 3520, 12960, 40320, 109584, 40320, 45, 249, 1350, 6750, 29880, 113400, 362880, 1026576, 362880, 55, 340, 2090, 12048

Offset: 1

Amarnath Murthy, Jul 10 2005

The first four columns (excluding the initial term of each) are A000217 (triangular numbers), A006527, A062026 and A062027. The first and third diagonals are both A000142 (factorials). The second diagonal is A000254.

Without the exception for k = n, a(n, n) would be n*n! (A001563(n)). For example, a(3, 3) would be 1*2*3 + 2*3*1 + 3*1*2 instead of 1*2*3. The author's original description did not mention the exception. I guess it didn't make sense to him to add n identical terms. - David Wasserman, Oct 01 2008

			a(5, 3) = 1*2*3 + 2*3*4 + 3*4*5 + 4*5*1 + 5*1*2 = 120.

Cf. A109877.

f(x, y) = if (x > y, x - y, x);
a(n, k) = if (n == k, n!, sum (i = 0, n - 1, prod (j = i + 1, i + k, f(j, n)))); \\ David Wasserman, Oct 01 2008

Edited and extended by David Wasserman, Oct 01 2008

A370108 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors in which the convex hull of a set of beads of any color A can be transformed by rotation into the convex hull of a set of beads of any other color B (n >= 1, k >= 1).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 0, 0, 0, 6, 2, 2, 0, 0, 10, 8, 6, 0, 0, 0, 15, 20, 18, 0, 2, 0, 0, 21, 40, 50, 0, 10, 0, 0, 0, 28, 70, 120, 24, 28, 0, 4, 0, 0, 36, 112, 252, 144, 60, 0, 12, 0, 0, 0, 45, 168, 476, 504, 230, 0, 54, 8, 4, 0

Offset: 1

Maxim Karimov and Vladislav Sulima, Feb 10 2024

It is assumed that all beads lie on a circle and distance between any two adjacent is the same.

			n\k| 1 2  3  4   5   6    7    8     9 ...
---+----------------------------------
 1 | 0 0  0  0   0   0    0    0     0 ... A000007
 2 | 0 1  3  6  10  15   21   28    36 ... A000217
 3 | 0 0  2  8  20  40   70  112   168 ... A007290
 4 | 0 2  6 18  50 120  252  476   828 ... A062026
 5 | 0 0  0  0  24 144  504 1344  3024 ... A059593
 6 | 0 2 10 28  60 230 1022 3640 10488
 7 | 0 0  0  0   0   0  720 5760 25920 ... A153760
 8 | 0 4 12 54 190 510 1134 7252 49284
 9 | 0 0  8 32  80 160  280  448 40992
...

Cf. A000007, A000013, A000217, A007290, A062026, A059593, A153760.

T(n,2) = A000013(ceiling(n/2)) * [n mod 2 == 0], where [] is the Iverson bracket.

For prime p, T(p,k) = (p-1)! * binomial(k,p).

cp's OEIS Frontend

A004255 n(n+1)(n^2 -3n + 6)/8.

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A109876 Triangle read by rows: a(n, n) = n! and for 1 <= k < n, a(n, k) = Sum_{i=0..n-1} Product_{j=i+1..i+k} f(j, n), where for x <= y, f(x, y) = x and for x > y, f(x, y) = x-y.

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A370108 Array read by antidiagonals: T(n,k) is the number of length n necklaces using at most k colors in which the convex hull of a set of beads of any color A can be transformed by rotation into the convex hull of a set of beads of any other color B (n >= 1, k >= 1).

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Examples

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Formula