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Number of partitions of 4n-1 into n nonnegative integers each no greater than 8.
(Formerly M3441 N1396)
1

%I M3441 N1396 #26 Jun 25 2023 02:54:50

%S 0,1,4,12,31,71,147,285,519,902,1502,2417,3768,5722,8481,12310,17528,

%T 24537,33814,45949,61629,81688,107089,138979,178669,227703,287828,

%U 361075,449731,556423,684089,836078,1016110,1228391,1477573,1768875,2108041,2501480

%N Number of partitions of 4n-1 into n nonnegative integers each no greater than 8.

%C In Cayley's terminology, this is the number of literal terms of degree n and of weight 4n-1 involving the letters a, b, c, d, e, f, g, h, i, having weights 0, 1, 2, 3, 4, 5, 6, 7, 8 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="/A001982/b001982.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_25">Index entries for linear recurrences with constant coefficients</a>, signature (1, 3, -1, -5, -2, 5, 4, -3, -3, 3, 2, -3, -3, 2, 3, -3, -3, 4, 5, -2, -5, -1, 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)(1-x^7z)(1-x^8z)), where w=4n-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%F G.f.: (x^14 +3*x^13 +5*x^12 +8*x^11 +13*x^10 +17*x^9 +19*x^8 +19*x^7 +19*x^6 +17*x^5 +13*x^4 +8*x^3 +5*x^2 +3*x+1)*x / ((x^4+x^3+x^2+x+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+x+1)^2 *(x+1)^3 *(x-1)^8). - _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)*(1-x^7*z)*(1-x^8*z)); n=400; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=4*d-1;print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%Y Cf. A001981.

%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