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 A003402 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)*(1-x^5)). (Formerly M1003) 13
 1, 1, 2, 4, 6, 9, 14, 19, 27, 37, 49, 64, 84, 106, 134, 168, 207, 253, 309, 371, 445, 530, 626, 736, 863, 1003, 1163, 1343, 1543, 1766, 2017, 2291, 2597, 2935, 3305, 3712, 4161, 4647, 5181, 5763, 6394, 7079, 7825, 8627, 9497, 10436, 11445, 12531, 13702, 14952 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Enumerates certain triangular arrays of integers. Also, Molien series for invariants of finite Coxeter group D_6 (bisected). The Molien series for the finite Coxeter group of type D_k (k >= 3) has G.f. = 1/Prod_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence. - N. J. A. Sloane, Jan 11 2016 REFERENCES J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Robert Israel, Table of n, a(n) for n = 0..10000 J. C. P. Miller, On the enumeration of partially ordered sets of integers, pp. 109-124 of T. P. McDonough and V. C. Mavron, editors, Combinatorics: Proceedings of the Fourth British Combinatorial Conference 1973. London Mathematical Society, Lecture Note Series, Number 13, Cambridge University Press, NY, 1974. The g.f. is in Eq. (8.1). Index entries for linear recurrences with constant coefficients, signature (1, 1, 1, -1, -2, -1, -1, 2, 2, 2, -1, -1, -2, -1, 1, 1, 1, -1). FORMULA a(n) = a(n-1) + b(n), b(n) = b(n-2) + c(n) - e(n), c(n) = c(n-3) + 2e(n), e(n) = e(n - 4) + f(n), f(n) = f(n - 5) + g(n), g(n) = g(n - 6), g(0) = 1, all functions are 0 for negative indexes. [From Miller paper.] - Sean A. Irvine, Apr 22 2015 a(n) = 1 + floor((7913/17280)*n + (13/96)*n^2 + (227/12960)*n^3 + (1/960)*n^4 + (1/43200)*n^5 + n/27*A079978(n) + n/128*(-1)^n). - Robert Israel, Apr 22 2015 MAPLE A079978:= n -> `if`(n mod 3 = 0, 1, 0): F:= n -> 1+floor((7913/17280)*n+(13/96)*n^2+(227/12960)*n^3+(1/960)*n^4+(1/43200)*n^5 + n/27*A079978(n) + n/128*(-1)^n): seq(F(n), n= 0..100); # Robert Israel, Apr 22 2015 MATHEMATICA CoefficientList[Series[1/((1 - x) (1 - x^2) (1 - x^3)^2*(1 - x^4) (1 - x^5)), {x, 0, 49}], x] (* Michael De Vlieger, Feb 21 2018 *) PROG (PARI) Vec(1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)*(1-x^5)) + O(x^50)) \\ Jinyuan Wang, Mar 10 2020 CROSSREFS Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775. Cf. A003403, A003404, A003405, A029073, A256975, A256976, A256977. Sequence in context: A024457 A117842 A067588 * A328863 A218004 A034748 Adjacent sequences:  A003399 A003400 A003401 * A003403 A003404 A003405 KEYWORD nonn AUTHOR EXTENSIONS Entry revised by N. J. A. Sloane, Apr 22 2015 STATUS approved

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Last modified July 13 22:21 EDT 2020. Contains 335716 sequences. (Running on oeis4.)