A001979 - OEIS

%I M3389 N1369 #36 Jun 25 2023 02:52:17

%S 1,1,4,10,24,49,94,169,289,468,734,1117,1656,2385,3370,4672,6375,8550,

%T 11322,14800,19138,24460,30982,38882,48417,59779,73316,89291,108108,

%U 130053,155646,185258,219489,258735,303748,355034,413442,479500,554256

%N Number of partitions of floor(7n/2) into n nonnegative integers each no more than 7.

%C Also, the dimension of the vector space of homogeneous covariants of degree n for the binary form of degree 7. To calculate the dimension one uses the Sylvester-Cayley formula. - _Leonid Bedratyuk_, Dec 06 2006

%C In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(7n/2) involving the letters a, b, c, d, e, f, g, h, having weights 0, 1, 2, 3, 4, 5, 6, 7 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).

%D Springer, T. A., Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, (1977).

%D Hilbert, D., Theory of algebraic invariants. Lectures. Cambridge University Press, (1993).

%H Alois P. Heinz, <a href="/A001979/b001979.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, 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_42">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 0, 1, -2, 2, -2, 2, -2, 1, 0, 0, 0, -2, 4, -4, 4, -3, 2, -1, 0, 1, -2, 3, -4, 4, -4, 2, 0, 0, 0, -1, 2, -2, 2, -2, 2, -1, 0, 1, -2, 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)), where w=floor(7n/2). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%F G.f.: -(x^34 -x^33 +3*x^32 +3*x^31 +7*x^30 +12*x^29 +16*x^28 +28*x^27 +33*x^26 +46*x^25 +56*x^24 +73*x^23 +83*x^22 +90*x^21 +106*x^20 +109*x^19 +121*x^18 +110*x^17 +121*x^16 +109*x^15 +106*x^14 +90*x^13 +83*x^12 +73*x^11 +56*x^10 +46*x^9 +33*x^8 +28*x^7 +16*x^6 +12*x^5 +7*x^4 +3*x^3 +3*x^2 -x+1) / ((x^4-x^2+1) *(x^4+x^3+x^2+x+1) *(x^4-x^3+x^2-x+1) *(x^4+1) *(x^2+x+1)^2 *(x^2-x+1)^2 *(x^2+1)^3 *(x+1)^5 *(x-1)^7). - _Alois P. Heinz_, Jul 25 2015

%p a(n+1) = subs({x=1},convert(series((product('1-x^i','i'=8..7+n)/product('1-x^k','k'=2..n)),x,trunc(7*n/2)+1),polynom)); # _Leonid Bedratyuk_, Dec 06 2006

%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)); n=450; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=floor(7*d/2);print1(polcoeff(polcoeff(p,w),d)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

%Y Cf. A001980.

%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