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 A302709 a(n) = Trinomial(2*n+1, 4) = (1/6)*n*(2*n + 1)*(2*n^2 + 9*n + 1), n >= 0. 1
 0, 6, 45, 161, 414, 880, 1651, 2835, 4556, 6954, 10185, 14421, 19850, 26676, 35119, 45415, 57816, 72590, 90021, 110409, 134070, 161336, 192555, 228091, 268324, 313650, 364481, 421245, 484386, 554364, 631655, 716751, 810160, 912406, 1024029, 1145585, 1277646, 1420800, 1575651, 1742819, 1922940 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The irregular triangle of trinomial coefficients is given in A027907. There the Comtet reference is given. 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). FORMULA a(n) = A027907(2*n+1, 4), n >= 0. a(n) = A027907(2*n+1, 2*(2*n-1)), for n >= 1 (symmetry). a(n) = binomial(2*n+1, 2) + (2*n+1)*binomial(2*n, 2) + binomial(2*n+1, 4) (from the trinomial definition) = (1/6)*n*(2*n + 1)*(2*n^2 + 9*n + 1). G.f.: x*(6 + 15*x - 4*x^2 - x^3)/(1 - x)^5. a(n) = (1/Pi)*Integral_{x=0..2} (1/sqrt(4 - x^2))*(x^2 - 1)^(2*n+1)*R(2*(2*n-3), 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+1, k = 4, rewritten with x = 2*cos(phi)]. MATHEMATICA CoefficientList[Series[x (6 + 15 x - 4 x^2 - x^3) / (1 - x)^5, {x, 0, 40}], x] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 6, 45, 161, 414}, 45] (* Vincenzo Librandi, Apr 20 2018 *) PROG (MAGMA) [(1/6)*n*(2*n+1)*(2*n^2+9*n+1): n in [0..50]]; // Vincenzo Librandi, Apr 20 2018 (PARI) a(n) = n*(2*n+1)*(2*n^2+9*n+1)/6; \\ Altug Alkan, Apr 20 2018 CROSSREFS Cf. A027907, A000384 (k=2), A030440 (k=3), A127672. Sequence in context: A204558 A123141 A122096 * A054563 A288835 A162230 Adjacent sequences:  A302706 A302707 A302708 * A302710 A302711 A302712 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Apr 19 2018 STATUS approved

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Last modified September 30 12:21 EDT 2020. Contains 337439 sequences. (Running on oeis4.)