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A040977 a(n) = binomial(n+5,5)*(n+3)/3. 33
1, 8, 35, 112, 294, 672, 1386, 2640, 4719, 8008, 13013, 20384, 30940, 45696, 65892, 93024, 128877, 175560, 235543, 311696, 407330, 526240, 672750, 851760, 1068795, 1330056, 1642473, 2013760, 2452472, 2968064, 3570952, 4272576, 5085465 (list; graph; refs; listen; history; text; internal format)
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

REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Jesus 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. 30 (2009), no. 1, 113-139. Also http://www.emis.ams.org/journals/JACO/Volume30_1/m6627810x2013373.fulltext.pdf. See page 138, n=4 entry in table.

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

Milan Janjic, Two Enumerative Functions

Index entries for sequences related to Chebyshev polynomials.

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

a(n) = A000579(n+5) + A000579(n+6). - R. J. Mathar, Nov 29 2015

Sum_{n>=0} 1/a(n) = 15*Pi^2 - 1175/8. - Jaume Oliver Lafont, Jul 11 2017

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

s1=s2=s3=s4=0; lst={}; Do[s1+=n^2; s2+=s1; s3+=s2; s4+=s3; AppendTo[lst, s4], {n, 0, 7!}]; lst (* Vladimir Joseph Stephan Orlovsky, Jan 15 2009 *)

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

Partial sums of A005585. Cf. A050486.

Cf. A000292, A133111, A133112.

Cf. A156925, A157703. - Johannes W. Meijer, Mar 07 2009

Sequence in context: A162211 A161717 A162494 * A266785 A267170 A266762

Adjacent sequences:  A040974 A040975 A040976 * A040978 A040979 A040980

KEYWORD

easy,nonn,changed

AUTHOR

Barry E. Williams, Dec 14 1999

STATUS

approved

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Last modified July 21 02:51 EDT 2017. Contains 289629 sequences.