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%I M3335 N1342 #43 Feb 04 2022 08:15:09
%S 1,1,4,8,18,32,58,94,151,227,338,480,676,920,1242,1636,2137,2739,3486,
%T 4370,5444,6698,8196,9926,11963,14293,17002,20076,23612,27594,32134,
%U 37212,42955,49341,56512,64444,73294,83036,93844,105690,118765,133037
%N Number of partitions of 3n into n parts from the set {0, 1, ..., 6} (repetitions admissible).
%C In Cayley's terminology, this is the number of literal terms of degree n and weight 3*n involving the letters a, b, c, d, e, f, g, having weights 0, 1, 2, 3, 4, 5, 6 respectively, a number which is also equal to the coefficient of x^(3n)z^n in the development of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008
%C a(0..5)=0; a(n) is the number of partitions of 3*(n+1) with 6 different numbers from the set {1,...,n}; the number of partitions of 3*(n+1)-C and 3*(n+1)+C are equal; example: n=8; 3*n+3=27; a(8)=4; (21,1), (22,1),(23,2), (24,2), (25,3), (26,3), (27,4), (28,3), (29,3), (30,2), (31,2),(32,1), (33,1). - _Paul Weisenhorn_, Jun 01 2009
%D A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
%D M. Jeger, Einfuehrung in die Kombinatorik, Band 2, Klett, 1975, pages 110- [From _Paul Weisenhorn_, Jun 01 2009]
%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 Alois P. Heinz, <a href="/A001977/b001977.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Cayley, <a href="http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;q1=second%20memoir%20on%20quantics;rgn=full%20text;cite1=cayley;cite1restrict=author;idno=ABS3153.0002.001;didno=ABS3153.0002.001;view=pdf;seq=00000289">Numerical tables supplementary to second memoir on quantics</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
%H A. Cayley, <a href="/A001971/a001971.pdf">Numerical tables supplementary to second memoir on quantics</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
%H Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/1901.08172">In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?</a>, arXiv:1901.08172 [math.CO], 2019.
%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-1,-1,-1,0,2,2,0,-1,-1,-1,-1,2,1,-1).
%F a(n) is the coefficient of x^(3*n+3) from the g.f. Product_{s=1..6} (x^s-x^(n+1))/(1-x^s). - _Paul Weisenhorn_, Jun 01 2009
%F G.f.: (x^10+x^8+3*x^7+4*x^6+4*x^5+4*x^4+3*x^3+x^2+1) / ((x^2+x+1) *(x^4+x^3+x^2+x+1) *(x^2+1) *(x+1)^3 *(x-1)^6). - _Alois P. Heinz_, Jul 26 2015
%t LinearRecurrence[{1, 2, -1, -1, -1, -1, 0, 2, 2, 0, -1, -1, -1, -1, 2, 1, -1}, {1, 1, 4, 8, 18, 32, 58, 94, 151, 227, 338, 480, 676, 920, 1242, 1636, 2137}, 100] (* _Jean-François Alcover_, Feb 25 2020 *)
%o (PARI) f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)*(1-x^6*z)); n=200; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(n=0,60,print1(polcoeff(polcoeff(p,3*n),n)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 16 2008
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