cp's OEIS Frontend

A054333 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).

%I A054333 #65 Jan 08 2025 12:49:49
%S A054333 1,11,65,275,935,2717,7007,16445,35750,72930,140998,260338,461890,
%T A054333 791350,1314610,2124694,3350479,5167525,7811375,11593725,16921905,
%U A054333 24322155,34467225,48208875,66615900,91018356,123058716,164750740
%N A054333 1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
%C A054333 If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-10) is the number of 10-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 08 2007
%C A054333 9-dimensional square numbers, eighth partial sums of binomial transform of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+8,i+8)*b(i)}, where b(i)=[1,2,0,0,0,...]. - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%C A054333 2*a(n) is number of ways to place 8 queens on an (n+8) X (n+8) chessboard so that they diagonally attack each other exactly 28 times. The maximal possible attack number, p=binomial(k,2) =28 for k=8 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
%D A054333 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
%D A054333 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
%D A054333 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%H A054333 G. C. Greubel, <a href="/A054333/b054333.txt">Table of n, a(n) for n = 0..5000</a>
%H A054333 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A054333 Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.
%H A054333 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 A054333 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%H A054333 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A054333 a(n) = (2*n+9)*binomial(n+8, 8)/9 = ((-1)^n)*A053120(2*n+9, 9)/2^8.
%F A054333 G.f.: (1+x)/(1-x)^10.
%F A054333 a(n) = 2*C(n+9, 9) - C(n+8, 8). - _Paul Barry_, Mar 04 2003
%F A054333 a(n) = C(n+8,8) + 2*C(n+8,9). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%F A054333 E.g.f.: (1/362880)*exp(x)*(362880 + 3628800*x + 7983360*x^2 + 6773760*x^3 + 2751840*x^4 + 592704*x^5 + 70560*x^6 + 4608*x^7 + 153*x^8 + 2*x^9). - _Stefano Spezia_, Dec 03 2018
%F A054333 From _Amiram Eldar_, Jan 26 2022: (Start)
%F A054333 Sum_{n>=0} 1/a(n) = 294912*log(2)/35 - 7153248/1225.
%F A054333 Sum_{n>=0} (-1)^n/a(n) = 73728*Pi/35 - 8105688/1225. (End)
%t A054333 LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {1, 11, 65, 275, 935, 2717, 7007, 16445, 35750, 72930}, 30] (* _Vincenzo Librandi_, Feb 14 2016 *)
%o A054333 (Magma) [Binomial(n+8,8)+2*Binomial(n+8,9): n in [0..40]]; // _Vincenzo Librandi_, Feb 14 2016
%o A054333 (PARI) vector(40, n, n--; (2*n+9)*binomial(n+8, 8)/9) \\ _G. C. Greubel_, Dec 02 2018
%o A054333 (Sage) [(2*n+9)*binomial(n+8, 8)/9 for n in range(40)] # _G. C. Greubel_, Dec 02 2018
%o A054333 (GAP)  List([0..30],n->(2*n+9)*Binomial(n+8,8)/9); # _Muniru A Asiru_, Dec 06 2018
%Y A054333 Partial sums of A053347. Cf. A053120, A000581.
%Y A054333 Cf. A005585, A040977, A050486, A053347. - _Vladimir Joseph Stephan Orlovsky_, Jan 15 2009
%Y A054333 Cf. A111125, fifth column (s=4, without leading zeros). - _Wolfdieter Lang_, Oct 18 2012
%K A054333 nonn,easy
%O A054333 0,2
%A A054333 _Barry E. Williams_, _Wolfdieter Lang_, Mar 15 2000