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A054333
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1/256 of tenth unsigned column of triangle A053120 (T-Chebyshev, rising powers, zeros omitted).
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14
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1, 11, 65, 275, 935, 2717, 7007, 16445, 35750, 72930, 140998, 260338, 461890, 791350, 1314610, 2124694, 3350479, 5167525, 7811375, 11593725, 16921905, 24322155, 34467225, 48208875, 66615900, 91018356, 123058716, 164750740
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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-10) is the number of 10-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 08 2007
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
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
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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.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 189, 194-196.
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 (10,-45,120,-210,252,-210,120,-45,10,-1).
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FORMULA
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a(n) = (2*n+9)*binomial(n+8, 8)/9 = ((-1)^n)*A053120(2*n+9, 9)/2^8.
G.f.: (1+x)/(1-x)^10.
a(n) = 2*C(n+9, 9) - C(n+8, 8). - Paul Barry, Mar 04 2003
a(n) = C(n+8,8) + 2*C(n+8,9). - Borislav St. Borisov (b.st.borisov(AT)abv.bg), Mar 05 2009
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
Sum_{n>=0} 1/a(n) = 294912*log(2)/35 - 7153248/1225.
Sum_{n>=0} (-1)^n/a(n) = 73728*Pi/35 - 8105688/1225. (End)
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MATHEMATICA
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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 *)
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PROG
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(Magma) [Binomial(n+8, 8)+2*Binomial(n+8, 9): n in [0..40]]; // Vincenzo Librandi, Feb 14 2016
(PARI) vector(40, n, n--; (2*n+9)*binomial(n+8, 8)/9) \\ G. C. Greubel, Dec 02 2018
(Sage) [(2*n+9)*binomial(n+8, 8)/9 for n in range(40)] # G. C. Greubel, Dec 02 2018
(GAP) List([0..30], n->(2*n+9)*Binomial(n+8, 8)/9); # Muniru A Asiru, Dec 06 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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