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

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

Offset: 0

Author

Barry E. Williams, Wolfdieter Lang, Mar 15 2000

Keywords

• nonn
• easy

Comments

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

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• 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].
• Milan Janjic, Two Enumerative Functions.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
• Index entries for sequences related to Chebyshev polynomials.

Crossrefs

Partial sums of A053347. Cf. A053120, A000581.

Cf. A005585, A040977, A050486, A053347. - Vladimir Joseph Stephan Orlovsky, Jan 15 2009

Cf. A111125, fifth column (s=4, without leading zeros). - Wolfdieter Lang, Oct 18 2012

Programs

GAP

List([0..30],n->(2*n+9)*Binomial(n+8,8)/9); # Muniru A Asiru, Dec 06 2018

Magma

[Binomial(n+8,8)+2*Binomial(n+8,9): n in [0..40]]; // Vincenzo Librandi, Feb 14 2016

Mathematica

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 *)

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

Formula

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

From Amiram Eldar, Jan 26 2022: (Start)

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)