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1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
14

%I #61 Jan 26 2022 02:30:29

%S 1,11,65,275,935,2717,7007,16445,35750,72930,140998,260338,461890,

%T 791350,1314610,2124694,3350479,5167525,7811375,11593725,16921905,

%U 24322155,34467225,48208875,66615900,91018356,123058716,164750740

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

%C 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 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 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 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 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

%H G. C. Greubel, <a href="/A054333/b054333.txt">Table of n, a(n) for n = 0..5000</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_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

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

%F a(n) = (2*n+9)*binomial(n+8, 8)/9 = ((-1)^n)*A053120(2*n+9, 9)/2^8.

%F G.f.: (1+x)/(1-x)^10.

%F a(n) = 2*C(n+9, 9) - C(n+8, 8). - _Paul Barry_, Mar 04 2003

%F a(n) = C(n+8,8) + 2*C(n+8,9). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009

%F 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 From _Amiram Eldar_, Jan 26 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 294912*log(2)/35 - 7153248/1225.

%F Sum_{n>=0} (-1)^n/a(n) = 73728*Pi/35 - 8105688/1225. (End)

%t 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 (Magma) [Binomial(n+8,8)+2*Binomial(n+8,9): n in [0..40]]; // _Vincenzo Librandi_, Feb 14 2016

%o (PARI) vector(40, n, n--; (2*n+9)*binomial(n+8, 8)/9) \\ _G. C. Greubel_, Dec 02 2018

%o (Sage) [(2*n+9)*binomial(n+8, 8)/9 for n in range(40)] # _G. C. Greubel_, Dec 02 2018

%o (GAP) List([0..30],n->(2*n+9)*Binomial(n+8,8)/9); # _Muniru A Asiru_, Dec 06 2018

%Y Partial sums of A053347. Cf. A053120, A000581.

%Y Cf. A005585, A040977, A050486, A053347. - _Vladimir Joseph Stephan Orlovsky_, Jan 15 2009

%Y Cf. A111125, fifth column (s=4, without leading zeros). - _Wolfdieter Lang_, Oct 18 2012

%K nonn,easy

%O 0,2

%A _Barry E. Williams_, _Wolfdieter Lang_, Mar 15 2000