A039623 a(n) = n^2*(n^2+3)/4.

1, 7, 27, 76, 175, 351, 637, 1072, 1701, 2575, 3751, 5292, 7267, 9751, 12825, 16576, 21097, 26487, 32851, 40300, 48951, 58927, 70357, 83376, 98125, 114751, 133407, 154252, 177451, 203175, 231601, 262912, 297297, 334951, 376075, 420876, 469567

Offset: 1

Author

Christian Meland (christian.meland(AT)pfi.no)

Keywords

• nonn
• easy
• nice

Comments

Previous definition was: Consider a figure like this <> (a squashed square, symmetric about both axes); each side is given 1 of n colors; a(n) = number of possibilities, allowing turning over.

Also number of 2 X 2 matrices with entries mod n, up to row and column permutation. Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively. - Vladeta Jovovic, Nov 04 2000

Also, if a 2-set Y and a 3-set Z are disjoint subsets of an n-set X then a(n-5) is the number of 6-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Jean-Paul Delahaye, Le miraculeux "lemme de Burnside", pp. 145-6 in 'Pour la Science' (French edition of 'Scientific American'), No. 350, December 2006, Paris.
• Milan Janjic, Two Enumerative Functions.
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A002724, A005353, A052271, A052272, A058001-A058004.

Row n=2 of A246106.

Programs

Magma

[n^2*(n^2+3)/4 : n in [1..50]]; // Wesley Ivan Hurt, Dec 26 2016

Maple

A039623:=n->n^2*(n^2+3)/4: seq(A039623(n), n=1..50); # Wesley Ivan Hurt, Dec 26 2016

Mathematica

Table[(n^2 (n^2+3))/4,{n,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,7,27,76,175},40] (* Harvey P. Dale, Oct 01 2011 *)

PARI

Vec((-1-2*x-2*x^2-x^3)/(x-1)^5 + O(x^50)) \\ Michel Marcus, Aug 23 2015

PARI

a(n) = (1/4)*n^2*(n^2+3); \\ Altug Alkan, Apr 16 2016

Formula

From Harvey P. Dale, Oct 01 2011: (Start)

G.f.: (1 + 2*x + 2*x^2 + x^3)/(1 - x)^5.

a(1)=1, a(2)=7, a(3)=27, a(4)=76, a(5)=175; for n>5, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)

E.g.f.: x*(4 + 10*x + 6*x^2 + x^3)*exp(x)/4. - Ilya Gutkovskiy, Apr 16 2016

a(n) = t(n-1)*t(n) + t(n-1) + t(n) where t=A000217. - J. M. Bergot, Apr 16 2016

a(n) = A000217(n)^2 - n*A000217(n-1). - Bruno Berselli, Feb 14 2017

a(n) = T(T(n-1)) + T(T(n)) where T(n) = A000217(n). - Charlie Marion, Feb 09 2023

Sum_{n>=1} 1/a(n) = 2*(1 + Pi^2 - sqrt(3)*Pi*coth(sqrt(3)*Pi))/9. - Amiram Eldar, Feb 13 2023

a(n) = binomial(n,2)*binomial(n+1,2) + n^2 = A006011(n) + A000290(n). - Detlef Meya, Nov 23 2023

Extensions

More terms from Sam Alexander

Simplified the definition. - N. J. A. Sloane, Apr 20 2016