A015401
Gaussian binomial coefficient [ n,10 ] for q=-12.
Original entry on oeis.org
1, 57154490053, 3563602618051323347605, 220521264778812882986788501660885, 13654753975171772337501943609360145428875733, 845462977543736084817433183822531039414960234418458069
Offset: 10
- J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
- M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13:
A015386,
A015388,
A015390,
A015391,
A015392,
A015393,
A015394,
A015397,
A015398,
A015399,
A015402.
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r:=10; q:=-12; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 04 2012
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Table[QBinomial[n, 10, -12], {n, 10, 20}] (* Vincenzo Librandi, Nov 05 2012 *)
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[gaussian_binomial(n,10,-12) for n in range(10,15)] # Zerinvary Lajos, May 25 2009
A015399
Gaussian binomial coefficient [ n,10 ] for q=-11.
Original entry on oeis.org
1, 23775972551, 621826557818118395106, 16116470915170412804822871108406, 418048302457998082359053173653182700919721, 10843028997901257369999365975865569183708813670389271
Offset: 10
- J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
- M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
Cf. Gaussian binomial coefficients [n, 10] for q = -2..-13:
A015386,
A015388,
A015390,
A015391,
A015392,
A015393,
A015394,
A015397,
A015398,
A015401,
A015402.
-
r:=10; q:=-11; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..20]]; // Vincenzo Librandi, Nov 05 2012
-
Table[QBinomial[n, 10, -11], {n, 10, 20}] (* Vincenzo Librandi, Nov 05 2012 *)
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[gaussian_binomial(n,10,-11) for n in range(10,16)] # Zerinvary Lajos, May 25 2009
A179897
a(n) = (n^(2*n+1) + 1) / (n+1).
Original entry on oeis.org
1, 1, 11, 547, 52429, 8138021, 1865813431, 593445188743, 250199979298361, 135085171767299209, 90909090909090909091, 74619186937936447687211, 73381705110822317661638341, 85180949465178001182799643437, 115244915978498073437814463065839, 179766618030828831251710653305053711
Offset: 0
Martin Saturka (martin(AT)saturka.net), Jul 31 2010
For n = 2, a(2) = 11 which is the arithmetic mean of {1/2, 1/2, 2^5} = 33 / 3 = 11. The geometric mean is 8^(1/3) = 2, i.e. both are integral.
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a(n) = (n^(2*n + 1) + 1)/(n + 1) \\ Andrew Howroyd, May 03 2023
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[(n**(2*n+1)+1)//(n+1) for n in range(1,11)]
Edited, a(0)=1 prepended and more terms from
Andrew Howroyd, May 03 2023
Comments