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Number of partitions of 3n-1 into n nonnegative integers each no more than 6.
(Formerly M2725 N1092)
1

%I M2725 N1092 #28 Jun 25 2023 02:50:30

%S 0,1,3,8,16,32,55,94,147,227,332,480,668,920,1232,1635,2124,2738,3470,

%T 4368,5424,6695,8172,9922,11934,14287,16968,20068,23572,27584,32087,

%U 37199,42901,49325,56450,64424,73223,83012,93764,105661,118674,133003,148616

%N Number of partitions of 3n-1 into n nonnegative integers each no more than 6.

%C In Cayley's terminology, this is the number of literal terms of degree n and of weight 3n-1 involving the letters a, b, c, d, e, f, g, having weights 0, 1, 2, 3, 4, 5, 6 respectively. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%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 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="/A001978/b001978.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 <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1, 3, -2, -4, 1, 3, -1, -1, 3, 1, -4, -2, 3, 1, -1).

%F Coefficient of x^w*z^n in the expansion of 1/((1-z)(1-xz)(1-x^2z)(1-x^3z)(1-x^4z)(1-x^5z)(1-x^6z)), where w=3n-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%F G.f.: (x^6 +2*x^5 +2*x^4 +x^3 +2*x^2 +2*x+1)*x / ((x^2+x+1) *(x^4+x^3+x^2+x+1) *(x+1)^3 *(x-1)^6). - _Alois P. Heinz_, Jul 25 2015

%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=400; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=3*d-1;print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%Y Cf. A001977.

%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 17 2008

%E a(0)=0 inserted by _Alois P. Heinz_, Jul 25 2015