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%I M3832 N1572 #35 Jan 25 2019 03:25:32
%S 1,1,5,13,33,73,151,289,526,910,1514,2430,3788,5744,8512,12346,17575,
%T 24591,33885,46029,61731,81805,107233,139143,178870,227930,288100,
%U 361384,450096,556834,684572,836618,1016737,1229093,1478379,1769773
%N Restricted partitions.
%C Number of partitions of 4n into up to 8 parts each no more than n; or partitions of 4n into up to n parts each no more than 8; or partitions of 5n into exactly n single-digit parts; or partitions of 4(n+2) into exactly 8 parts each no more than n+1; or partitions of 4(n+9) into exactly 8 distinct parts each no more than n+8; etc. Points lie on 252 different septics with the pattern repeating every 420 points, amounting to 4 sets of parallel septics depending on whether n mod 6 is in {0}, {1,5}, {2,4} or {3}.
%C Also, the dimension of the vector space of homogeneous covariants of degree n for the binary form of degree 8. - _Leonid Bedratyuk_, Dec 06 2006
%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 Hilbert, D., Theory of algebraic invariants. Lectures. Cambridge University Press, (1993).
%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).
%H Alois P. Heinz, <a href="/A001981/b001981.txt">Table of n, a(n) for n = 0..1000</a>
%H Henry Bottomley, <a href="http://www.se16.info/js/partitions.htm">Partition and composition calculator</a>.
%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.
%F a(n) =A067059(n, 8) =A067059(8, n) =(1/152409600) * (1812n^7 + 57078n^6 + 748314n^5 + 5278770n^4 + 21727272n^3 + 52982181n^2 + 77609245n + 66220839 + (297675n^2+2679075n+27088425)*(1, -1)pcr(n, 2) + (1254400*n+5644800)*(2, -1, -1)pcr(n, 3) + 9408000*(0, -1, 1)pcr(n, 3) + 4762800*(1, 1, -1, -1)pcr(n, 4) + 24385536*(1, -1, 0, 0, 0)pcr(n, 5) + 6220800(3, -1, 2, -2, 1, -3, 0)pcr(n, 7)) where for example (0, -1, 1)pcr(n, 3) means the value 0 if n mod 3 = 0, the value -1 if n mod 3 = 1 and the value 1 if n mod 3 = 2. - _Henry Bottomley_, Jul 19 2003
%e a(3)=13 since partitions of 12 into up to 8 parts each no more than 3 are 3+3+3+3 = 3+3+3+2+1 = 3+3+3+1+1+1 = 3+3+2+2+2 = 3+3+2+2+1+1 = 3+3+2+1+1+1+1 = 3+3+1+1+1+1+1+1 = 3+2+2+2+2+1 = 3+2+2+2+1+1+1 = 3+2+2+1+1+1+1+1 = 2+2+2+2+2+2 = 2+2+2+2+2+1+1 = 2+2+2+2+1+1+1+1; or equivalently partitions of 15 into exactly 3 single-digit numbers are 9+5+1 = 9+4+2 = 9+3+3 = 8+6+1 = 8+5+2 = 8+4+3 = 7+7+1 = 7+6+2 = 7+5+3 = 7+4+4 = 6+6+3 = 6+5+4 =5+5+5.
%p a:= n-> subs({x=1}, convert(series((product('1-x^i', 'i'=9..8+n)/ product('1-x^k', 'k'=2..n)), x, 4*n+1), polynom)): seq (a(n), n=0..40); # _Leonid Bedratyuk_, Dec 06 2006
%t a[n_] := Length[IntegerPartitions[4*n, 8, Range[n]]]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Mar 17 2014 *)
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E Edited by _Henry Bottomley_, Jul 19 2003
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