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A054334
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1/512 of 11th unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
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15
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1, 12, 77, 352, 1287, 4004, 11011, 27456, 63206, 136136, 277134, 537472, 999362, 1790712, 3105322, 5230016, 8580495, 13748020, 21559395, 33153120, 50075025, 74397180, 108864405, 157073280, 223689180, 314707536, 437766252, 602516992, 821063892, 1108479152
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OFFSET
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0,2
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COMMENTS
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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
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]
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
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REFERENCES
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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.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
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FORMULA
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a(n) = (2*n+10)*binomial(n+9, 9)/10 = ((-1)^n)*A053120(2*n+10, 10)/2^9.
G.f.: (1+x)/(1-x)^11.
a(n) = 2*binomial(n+10, 10) - binomial(n+9, 9). - Paul Barry, Mar 04 2003
a(n) = binomial(n+9,9) + 2*binomial(n+9,10). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
Sum_{n>=0} 1/a(n) = 525*Pi^2 - 1160419/224.
Sum_{n>=0} (-1)^n/a(n) = 525*Pi^2/2 - 82875/32. (End)
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MATHEMATICA
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PROG
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(PARI) vector(40, n, n--; (2*n+10)*binomial(n+9, 9)/10) \\ G. C. Greubel, Dec 02 2018
(Magma) [(2*n+10)*Binomial(n+9, 9)/10: n in [0..40]]; // G. C. Greubel, Dec 02 2018
(Sage) [(2*n+10)*binomial(n+9, 9)/10 for n in range(40)] # G. C. Greubel, Dec 02 2018
(GAP) List([0..30], n -> (2*n+10)*Binomial(n+9, 9)/10); # G. C. Greubel, Dec 02 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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