A050486 a(n) = binomial(n+6,6)*(2n+7)/7.

1, 9, 44, 156, 450, 1122, 2508, 5148, 9867, 17875, 30888, 51272, 82212, 127908, 193800, 286824, 415701, 591261, 826804, 1138500, 1545830, 2072070, 2744820, 3596580, 4665375, 5995431, 7637904, 9651664, 12104136, 15072200, 18643152, 22915728, 28001193

Offset: 0

Author

Barry E. Williams, Dec 26 1999

Keywords

• nonn
• easy

Comments

If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-8) is the number of 8-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007

7-dimensional square numbers, sixth partial sums of binomial transform of [1,2,0,0,0,...]. a(n) = Sum_{i=0..n} C(n+6,i+6)*b(i), where b(i) = [1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

2*a(n) is number of ways to place 6 queens on an (n+6) X (n+6) chessboard so that they diagonally attack each other exactly 15 times. The maximal possible attack number, p=binomial(k,2)=15 for k=6 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

Coefficients in the terminating series identity 1 - 9*n/(n + 8) + 44*n*(n - 1)/((n + 8)*(n + 9)) - 156*n*(n - 1)*(n - 2)/((n + 8)*(n + 9)*(n + 10)) + ... = 0 for n = 1,2,3,.... Cf. A005585 and A053347. - Peter Bala, Feb 18 2019

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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 pp. 15, 17.
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Partial sums of A040977, A005585.

Fourth column (s=3, without leading zeros) of A111125. - Wolfdieter Lang, Oct 18 2012

Cf. A084960 (unsigned fourth column divided by 64). - Wolfdieter Lang, Aug 04 2014

Cf. A053120, A084930.

Programs

Magma

[Binomial(n+6, 6) + 2*Binomial(n+6, 7): n in [0..35]]; // Vincenzo Librandi, Jun 09 2013

Maple

A050486:=n->binomial(n+6,6)*(2*n+7)/7: seq(A050486(n), n=0..50); # Wesley Ivan Hurt, Jan 01 2016

Mathematica

CoefficientList[Series[(1 + x) / (1 - x)^8, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
Table[SeriesCoefficient[(1 + x)/(1 - x)^8, {x, 0, n}], {n, 0, 28}] (* or *)
Table[Binomial[n + 6, 6] (2 n + 7)/7, {n, 0, 30}] (* Michael De Vlieger, Dec 31 2015 *)

PARI

a(n)=binomial(n+6,6)*(2*n+7)/7 \\ Charles R Greathouse IV, Sep 24 2015

Python

A050486_list, m = [], [2]+[1]*7
for _ in range(10**2):
    A050486_list.append(m[-1])
    for i in range(7):
        m[i+1] += m[i] # Chai Wah Wu, Jan 24 2016

Formula

a(n) = (-1)^n*A053120(2*n+7, 7)/64 (1/64 of eighth unsigned column of Chebyshev T-triangle, zeros omitted).

G.f.: (1+x)/(1-x)^8.

a(n) = 2*C(n+7, 7)-C(n+6, 6). - Paul Barry, Mar 04 2003

a(n) = C(n+6,6)+2*C(n+6,7). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

a(n) = (-1)^n*A084930(n+3, 3)/64. Compare with the first line above. - Wolfdieter Lang, Aug 04 2014

a(n) = 8*a(n-1)-28*a(n-2)+56*a(n-3)-70*a(n-4)+56*a(n-5)-28*a(n-6)+8*a(n-7)-a(n-8) for n>7. - Wesley Ivan Hurt, Jan 01 2016

From Amiram Eldar, Jan 25 2022: (Start)

Sum_{n>=0} 1/a(n) = 24871/25 - 7168*log(2)/5.

Sum_{n>=0} (-1)^n/a(n) = 1792*Pi/5 - 28126/25. (End)