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%I #36 Jan 26 2022 02:30:23
%S 1,12,77,352,1287,4004,11011,27456,63206,136136,277134,537472,999362,
%T 1790712,3105322,5230016,8580495,13748020,21559395,33153120,50075025,
%U 74397180,108864405,157073280,223689180,314707536,437766252,602516992,821063892,1108479152
%N 1/512 of 11th unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
%C Partial sums of A054333.
%C If a 2-set Y and an (n-3)-set Z are disjoint subsets of an n-set X then a(n-11) is the number of 11-subsets of X intersecting both Y and Z. - _Milan Janjic_, Sep 08 2007
%C 10-dimensional square numbers, ninth partial sums of binomial transform of [1,2,0,0,0,...]. a(n)=sum{i=0,n,C(n+9,i+9)*b(i)}, where b(i)=[1,2,0,0,0,...]. [From Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009]
%C 2*a(n) is number of ways to place 9 queens on an (n+9) X (n+9) chessboard so that they diagonally attack each other exactly 36 times. The maximal possible attack number, p=binomial(k,2)=36 for k=9 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 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 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%H T. D. Noe, <a href="/A054334/b054334.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F a(n) = (2*n+10)*binomial(n+9, 9)/10 = ((-1)^n)*A053120(2*n+10, 10)/2^9.
%F G.f.: (1+x)/(1-x)^11.
%F a(n) = 2*binomial(n+10, 10) - binomial(n+9, 9). - _Paul Barry_, Mar 04 2003
%F a(n) = binomial(n+9,9) + 2*binomial(n+9,10). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
%F a(n) = binomial(n+9,9)*(n+5)/5. - _Antal Pinter_, Dec 27 2015
%F From _Amiram Eldar_, Jan 26 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 525*Pi^2 - 1160419/224.
%F Sum_{n>=0} (-1)^n/a(n) = 525*Pi^2/2 - 82875/32. (End)
%t Table[(2*n + 10)*Binomial[n + 9, 9]/10, {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jan 15 2009 *)
%o (PARI) vector(40, n, n--; (2*n+10)*binomial(n+9, 9)/10) \\ _G. C. Greubel_, Dec 02 2018
%o (Magma) [(2*n+10)*Binomial(n+9, 9)/10: n in [0..40]]; // _G. C. Greubel_, Dec 02 2018
%o (Sage) [(2*n+10)*binomial(n+9, 9)/10 for n in range(40)] # _G. C. Greubel_, Dec 02 2018
%o (GAP) List([0..30], n -> (2*n+10)*Binomial(n+9, 9)/10); # _G. C. Greubel_, Dec 02 2018
%Y Cf. A053120, A054333.
%Y Cf. A005585, A040977, A050486, A053347, A054333.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_
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