cp's OEIS Frontend

A053347 a(n) = binomial(n+7, 7)*(n+4)/4.

%I A053347 #73 Jan 08 2025 12:20:00
%S A053347 1,10,54,210,660,1782,4290,9438,19305,37180,68068,119340,201552,
%T A053347 329460,523260,810084,1225785,1817046,2643850,3782350,5328180,7400250,
%U A053347 10145070,13741650,18407025,24402456,32040360,41692024,53796160
%N A053347 a(n) = binomial(n+7, 7)*(n+4)/4.
%C A053347 If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-9) is the number of 9-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 08 2007
%C A053347 8-dimensional square numbers, seventh partial sums of binomial transform of [1, 2, 0, 0, 0, ...]. a(n) = sum{i=0,n,C(n+7, i+7)*b(i)}, where b(i) = [1, 2, 0, 0, 0, ...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%C A053347 2*a(n) is number of ways to place 7 queens on an (n+7) X (n+7) chessboard so that they diagonally attack each other exactly 21 times. The maximal possible attack number, p=binomial(k,2)=21 for k=7 queens, is achievable only when all queens are on the same diagonal. In graph-theory representation they thus form the corresponding complete graph. - _Antal Pinter_, Dec 27 2015
%C A053347 Coefficients in the terminating series identity 1 - 10*n/(n + 9) + 54*n*(n - 1)/((n + 9)*(n + 10)) - 210*n*(n - 1)*(n - 2)/((n + 9)*(n + 10)*(n + 11)) + ... = 0 for n = 1,2,3,.... Cf. A050486. - _Peter Bala_, Feb 18 2019
%D A053347 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%H A053347 Vincenzo Librandi, <a href="/A053347/b053347.txt">Table of n, a(n) for n = 0..1000</a>
%H A053347 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>
%H A053347 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 p. 15.
%H A053347 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%H A053347 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A053347 a(n) = ((-1)^n)*A053120(2*n+8, 8)/2^7 (1/128 of ninth unsigned column of Chebyshev T-triangle, zeros omitted).
%F A053347 G.f.: (1+x)/(1-x)^9.
%F A053347 a(n) = 2*C(n+8, 8) - C(n+7, 7). - _Paul Barry_, Mar 04 2003
%F A053347 a(n) = A027803(n-3)/35 = C(n+4, n)*C(n+7, 4)/35. - _Zerinvary Lajos_, May 25 2005
%F A053347 a(n) = C(n+7, 7) + 2*C(n+7, 8). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%F A053347 a(n) = (n^8 + 32*n^7 + 434*n^6 + 3248*n^5 + 14609*n^4 + 40208*n^3 + 65596*n^2 + 57312*n + 20160)/20160. - _Chai Wah Wu_, Jan 24 2016
%F A053347 Sum_{n>=0} 1/a(n) = 41503/45 - 280/3*Pi^2. - _Jaume Oliver Lafont_, Jul 17 2017
%F A053347 Sum_{n>=0} (-1)^n/a(n) = 140*Pi^2/3 - 1379/3. - _Amiram Eldar_, Jan 25 2022
%p A053347 A053347:=n->binomial(n+7,7)*(n+4)/4: seq(A053347(n), n=0..50); # _Wesley Ivan Hurt_, Jul 16 2017
%t A053347 s1=s2=s3=s4=s5=s6=0; lst={}; Do[s1+=n^2; s2+=s1; s3+=s2; s4+=s3; s5+=s4; s6+=s5; AppendTo[lst,s6],{n,0,7!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jan 15 2009 *)
%t A053347 CoefficientList[Series[(1 + x) / (1 - x)^9, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%t A053347 Table[SeriesCoefficient[(1 + x)/(1 - x)^9, {x, 0, n}], {n, 0, 28}] (* or *)
%t A053347 Table[Binomial[n + 7, 7] (n + 4)/4, {n, 0, 28}] (* _Michael De Vlieger_, Dec 31 2015 *)
%o A053347 (PARI) a(n)=binomial(n+7,7)*(n+4)/4 \\ _Charles R Greathouse IV_, Jun 10 2011
%o A053347 (Magma) [Binomial(n+7,7)+2*Binomial(n+7,8): n in [0..35]]; // _Vincenzo Librandi_, Jun 09 2013
%o A053347 (Python)
%o A053347 A053347_list, m = [], [2]+[1]*8
%o A053347 for _ in range(10**2):
%o A053347     A053347_list.append(m[-1])
%o A053347     print(m[-1])
%o A053347     for i in range(8):
%o A053347         m[i+1] += m[i] # _Chai Wah Wu_, Jan 24 2016
%Y A053347 Partial sums of A050486.
%Y A053347 Cf. A005585, A040977.
%K A053347 easy,nonn
%O A053347 0,2
%A A053347 _Barry E. Williams_, Jan 06 2000