cp's OEIS Frontend

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

%I A050486 #77 Aug 11 2025 01:54:12
%S A050486 1,9,44,156,450,1122,2508,5148,9867,17875,30888,51272,82212,127908,
%T A050486 193800,286824,415701,591261,826804,1138500,1545830,2072070,2744820,
%U A050486 3596580,4665375,5995431,7637904,9651664,12104136,15072200,18643152,22915728,28001193
%N A050486 a(n) = binomial(n+6,6)*(2n+7)/7.
%C A050486 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
%C A050486 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
%C A050486 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
%C A050486 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
%D A050486 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A050486 Vincenzo Librandi, <a href="/A050486/b050486.txt">Table of n, a(n) for n = 0..1000</a>
%H A050486 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A050486 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See pp. 15, 17.
%H A050486 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%H A050486 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A050486 a(n) = (-1)^n*A053120(2*n+7, 7)/64 (1/64 of eighth unsigned column of Chebyshev T-triangle, zeros omitted).
%F A050486 G.f.: (1+x)/(1-x)^8.
%F A050486 a(n) = 2*C(n+7, 7)-C(n+6, 6). - _Paul Barry_, Mar 04 2003
%F A050486 a(n) = C(n+6,6)+2*C(n+6,7). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%F A050486 a(n) = (-1)^n*A084930(n+3, 3)/64. Compare with the first line above. - _Wolfdieter Lang_, Aug 04 2014
%F A050486 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
%F A050486 From _Amiram Eldar_, Jan 25 2022: (Start)
%F A050486 Sum_{n>=0} 1/a(n) = 24871/25 - 7168*log(2)/5.
%F A050486 Sum_{n>=0} (-1)^n/a(n) = 1792*Pi/5 - 28126/25. (End)
%p A050486 A050486:=n->binomial(n+6,6)*(2*n+7)/7: seq(A050486(n), n=0..50); # _Wesley Ivan Hurt_, Jan 01 2016
%t A050486 CoefficientList[Series[(1 + x) / (1 - x)^8, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%t A050486 Table[SeriesCoefficient[(1 + x)/(1 - x)^8, {x, 0, n}], {n, 0, 28}] (* or *)
%t A050486 Table[Binomial[n + 6, 6] (2 n + 7)/7, {n, 0, 30}] (* _Michael De Vlieger_, Dec 31 2015 *)
%o A050486 (Magma) [Binomial(n+6, 6) + 2*Binomial(n+6, 7): n in [0..35]]; // _Vincenzo Librandi_, Jun 09 2013
%o A050486 (PARI) a(n)=binomial(n+6,6)*(2*n+7)/7 \\ _Charles R Greathouse IV_, Sep 24 2015
%o A050486 (Python)
%o A050486 A050486_list, m = [], [2]+[1]*7
%o A050486 for _ in range(10**2):
%o A050486     A050486_list.append(m[-1])
%o A050486     for i in range(7):
%o A050486         m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%Y A050486 Partial sums of A040977, A005585.
%Y A050486 Fourth column (s=3, without leading zeros) of A111125. - _Wolfdieter Lang_, Oct 18 2012
%Y A050486 Cf. A084960 (unsigned fourth column divided by 64). - _Wolfdieter Lang_, Aug 04 2014
%Y A050486 Cf. A053120, A084930.
%K A050486 nonn,easy
%O A050486 0,2
%A A050486 _Barry E. Williams_, Dec 26 1999