A302710 a(n) = trinomial(2*n, 4) = (1/6)*n*(2*n - 1)*(2*n^2 + 7*n - 3).
0, 1, 19, 90, 266, 615, 1221, 2184, 3620, 5661, 8455, 12166, 16974, 23075, 30681, 40020, 51336, 64889, 80955, 99826, 121810, 147231, 176429, 209760, 247596, 290325, 338351, 392094, 451990, 518491, 592065, 673196, 762384, 860145, 967011, 1083530, 1210266, 1347799, 1496725, 1657656, 1831220
Offset: 0
References
- L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 77-78. (In the integral formula on p. 77 a left bracket is missing for the cosine argument.)
Links
- Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5, 1).
Programs
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Magma
[(1/6)*n*(2*n-1)*(2*n^2+7*n-3): n in [0..40]]; // Vincenzo Librandi, Apr 28 2018
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Mathematica
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 19, 90, 266}, 41] (* Vincenzo Librandi, Feb 28 2018 *) -
PARI
a(n) = n*(2*n-1)*(2*n^2+7*n-3)/6; \\ Altug Alkan, May 01 2018
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PARI
a(n) = polcoeff((1 + x + x^2)^(2*n), 4); \\ Michel Marcus, May 04 2018
Formula
a(n) = binomial(2*n, 2) + (2*n)*binomial(2*n-1, 2) + binomial(2*n, 4)(from the trinomial definition) = (1/6)*n*(2*n - 1)*(2*n^2 + 7*n - 3).
G.f.: x*(1 + 14*x + 5*x^2 - 4*x^3)/(1 - x)^5.
a(n) = (1/Pi)*Integral_{x=0..2} (1/sqrt(4 - x^2))*(x^2 - 1)^(2*n)*R(4*(n-2), x), n >= 0, with the R polynomial coefficients given in A127672. Note that R(-n, x) = R(n, x) [Comtet, p. 77, the integral formula for q = 3, n -> 2*n, k = 4, rewritten with x = 2*cos(phi)]. For the odd numbered rows see A302709.
Comments