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A040977
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a(n) = binomial(n+5,5)*(n+3)/3.
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37
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-16.
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LINKS
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FORMULA
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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) = 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
Sum_{n>=0} (-1)^n/a(n) = 15*Pi^2/2 - 585/8. - Amiram Eldar, Jan 24 2022
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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