OFFSET
0,2
COMMENTS
Sequence is n^2*(n^2-1)*(n^2-4)/360 if offset 3.
If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-7) is the number of 7-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
6-dimensional square numbers, fifth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} binomial(n+5,i+5)*b(i), where b(i) = [1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
Sequence of the absolute values of the z^2 coefficients divided by 5 of the polynomials in the GF2 denominators of A156925. See A157703 for background information. - Johannes W. Meijer, Mar 07 2009
2*a(n) is number of ways to place 5 queens on an (n+5) X (n+5) chessboard so that they diagonally attack each other exactly 10 times. The maximal possible attack number, p=binomial(k,2)=10 for k=5 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form a corresponding complete graph. - Antal Pinter, Dec 27 2015
Ehrhart polynomial for the Chan-Robbins-Yuen polytope CRY_4. [De Loera et al.] - N. J. A. Sloane, Apr 16 2016
Coefficients in the terminating series identity 1 - 8*n/(n + 7) + 35*n*(n - 1)/((n + 7)*(n + 8)) - 112*n*(n - 1)*(n - 2)/((n + 7)*(n + 8)*(n + 9)) + ... = 0 for n = 1,2,3,.... Cf. A005585 and A050486. - Peter Bala, Feb 18 2019
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 216.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jesús A. De Loera, Fu Liu and Ruriko Yoshida, A generating function for all semi-magic squares and the volume of the Birkhoff polytope, J. Algebraic Combin., Vol. 30, No. 1 (2009), pp. 113-139. See page 138, n=4 entry in table.
Milan Janjic, Two Enumerative Functions.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = (-1)^n*A053120(2*n+6, 6)/32, (1/32 of seventh unsigned column of Chebyshev T-triangle, zeros omitted).
G.f.: (1+x)/(1-x)^7.
a(n-3) = Sum_{i+j+k=n} i*j*k^2. - Benoit Cloitre, Nov 01 2002
a(n) = 2*binomial(n+6, 6) - binomial(n+5, 5). - Paul Barry, Mar 04 2003
a(n-3) = 1/(1!*2!*3!)*Sum_{1 <= x_1, x_2, x_3 <= n} |det V(x_1,x_2,x_3)| = 1/12*Sum_{1 <= i,j,k <= n} |(i-j)(i-k)(j-k)|, where V(x_1,x_2,x_3) is the Vandermonde matrix of order 3. - Peter Bala, Sep 13 2007
a(n) = binomial(n+5,5) + 2*binomial(n+5,6). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
a(n) = (n+1)*(n+2)*(n+3)^2*(n+4)*(n+5)/360. - Wesley Ivan Hurt, May 05 2015
Sum_{n>=0} 1/a(n) = 15*Pi^2 - 1175/8. - Jaume Oliver Lafont, Jul 11 2017
Sum_{n>=0} (-1)^n/a(n) = 15*Pi^2/2 - 585/8. - Amiram Eldar, Jan 24 2022
E.g.f.: exp(x)*(360 + 2520*x + 3600*x^2 + 1800*x^3 + 375*x^4 + 33*x^5 + x^6)/360. - Stefano Spezia, Oct 30 2025
MAPLE
with(combinat); A040977 := n->binomial(n+5, 5)*(n+3)/3;
a:=n->(sum((numbcomp(n, 6)), j=4..n))/3:seq(a(n), n=6..38); # Zerinvary Lajos, Aug 26 2008
nmax:=34; for n from 0 to nmax do fz(n):=product((1-m*z)^(n+1-m), m=1..n); c(n):= abs(coeff(fz(n), z, 2))/5; end do: a:=n-> c(n): seq(a(n), n=2..nmax); # Johannes W. Meijer, Mar 07 2009
MATHEMATICA
CoefficientList[Series[(1 + x) / (1 - x)^7, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {1, 8, 35, 112, 294, 672, 1386}, 40] (* Harvey P. Dale, Feb 20 2016 *)
PROG
(Magma) [Binomial(n+5, 5) + 2*Binomial(n+5, 6): n in [0..35]]; // Vincenzo Librandi, Jun 09 2013
(PARI) vector(20, n, n--; 2*binomial(n+6, 6)-binomial(n+5, 5)) \\ Derek Orr, May 05 2015
(PARI) Vec((1+x)/(1-x)^7 + O(x^100)) \\ Altug Alkan, Nov 29 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Dec 14 1999
STATUS
approved