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a(n) = binomial(n+6,6)*(2n+7)/7.
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%I #66 Sep 08 2022 08:44:58

%S 1,9,44,156,450,1122,2508,5148,9867,17875,30888,51272,82212,127908,

%T 193800,286824,415701,591261,826804,1138500,1545830,2072070,2744820,

%U 3596580,4665375,5995431,7637904,9651664,12104136,15072200,18643152,22915728,28001193

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

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

%H Vincenzo Librandi, <a href="/A050486/b050486.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>

%H <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 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.

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

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

%F a(n) = 2*C(n+7, 7)-C(n+6, 6). - _Paul Barry_, Mar 04 2003

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

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

%F 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 From _Amiram Eldar_, Jan 25 2022: (Start)

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

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

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

%t s1=s2=s3=s4=s5=0; lst={}; Do[s1+=n^2; s2+=s1; s3+=s2; s4+=s3; s5+=s4; AppendTo[lst,s5],{n,0,7!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 15 2009 *)

%t CoefficientList[Series[(1 + x) / (1 - x)^8, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 )

%t Table[SeriesCoefficient[(1 + x)/(1 - x)^8, {x, 0, n}], {n, 0, 28}] (* or *)

%t Table[Binomial[n + 6, 6] (2 n + 7)/7, {n, 0, 30}] (* _Michael De Vlieger_, Dec 31 2015 *)

%o (Magma) [Binomial(n+6, 6) + 2*Binomial(n+6, 7): n in [0..35]]; // _Vincenzo Librandi_, Jun 09 2013

%o (PARI) a(n)=binomial(n+6,6)*(2*n+7)/7 \\ _Charles R Greathouse IV_, Sep 24 2015

%o (Python)

%o A050486_list, m = [], [2]+[1]*7

%o for _ in range(10**2):

%o A050486_list.append(m[-1])

%o for i in range(7):

%o m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016

%Y Partial sums of A040977, A005585.

%Y Fourth column (s=3, without leading zeros) of A111125. - _Wolfdieter Lang_, Oct 18 2012

%Y Cf. A084960 (unsigned fourth column divided by 64). - _Wolfdieter Lang_, Aug 04 2014

%Y Cf. A053120, A084930.

%K nonn,easy

%O 0,2

%A _Barry E. Williams_, Dec 26 1999