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 A001996 Number of partitions of n into parts 2, 3, 4, 5, 6, 7. (Formerly M0306 N0112) 12

%I M0306 N0112

%S 1,0,1,1,2,2,4,4,6,7,10,11,16,17,23,26,33,37,47,52,64,72,86,96,115,

%T 127,149,166,192,212,245,269,307,338,382,419,472,515,576,629,699,760,

%U 843,913,1007,1091,1197,1293,1416,1525,1663,1790,1945,2088,2265,2426

%N Number of partitions of n into parts 2, 3, 4, 5, 6, 7.

%C Also, Molien series for invariants of finite Coxeter group A_6. The Molien series for the finite Coxeter group of type A_k (k >= 1) has G.f. = 1/Prod_{i=2..k+1} (1-x^i). - _N. J. A. Sloane_, Jan 11 2016

%C Cayley tabulates the coefficients in the expansion of H = 1 / ((1 - x^2) * (1 - x^4) * ... * (1 - x^14)) with even indices 0, 2, ..., 142.

%D A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001996/b001996.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Cayley, <a href="http://www.jstor.org/stable/2369198">Calculation of the minimum N.G.F. of the binary seventhic</a>, American Journal of Mathematics, 2 (1879), pp.71-84. See pp.77-78.

%H A. Cayley, <a href="/A001993/a001993.pdf">Calculation of the minimum N.G.F. of the binary seventhic</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]

%H <a href="/index/Rec#order_27">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 1, 1, 0, 0, -1, -2, -2, -1, 0, 2, 2, 2, 2, 0, -1, -2, -2, -1, 0, 0, 1, 1, 1, 0, -1).

%F G.f.: 1/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^7)).

%F Euler transform of length 7 sequence [ 0, 1, 1, 1, 1, 1, 1]. - _Michael Somos_, Apr 23 2014

%e G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 4*x^6 + 4*x^7 + 6*x^8 + 7*x^9 + ...

%e G.f. = 1 + q^2 + q^6 + 2*q^8 + 2*q^10 + 4*q^12 + 4*q^14 + 6*q^16 + ...

%t nn = 102; t = CoefficientList[Series[1/((1 - x^4)*(1 - x^6)*(1 - x^8)*(1 - x^10)*(1 - x^12)*(1 - x^14)), {x, 0, nn}], x]; t = Take[t, {1, nn, 2}]

%Y Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.

%K nonn,easy

%O 0,5

%A _N. J. A. Sloane_.

%E More terms from _James A. Sellers_, Feb 09 2000

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Last modified February 21 20:21 EST 2020. Contains 332111 sequences. (Running on oeis4.)