A001981 Restricted partitions.

1, 1, 5, 13, 33, 73, 151, 289, 526, 910, 1514, 2430, 3788, 5744, 8512, 12346, 17575, 24591, 33885, 46029, 61731, 81805, 107233, 139143, 178870, 227930, 288100, 361384, 450096, 556834, 684572, 836618, 1016737, 1229093, 1478379, 1769773

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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}.

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

Examples

			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.
		

References

• 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.
• Hilbert, D., Theory of algebraic invariants. Lectures. Cambridge University Press, (1993).
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Springer, T.A., Invariant theory, Lecture Notes in Mathematics, 585, Springer-Verlag, (1977).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Henry Bottomley, Partition and composition calculator.
• 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. [Annotated scanned copy]
• Shalosh B. Ekhad, Doron Zeilberger, In How many ways can I carry a total of n coins in my two pockets, and have the same amount in both pockets?, arXiv:1901.08172 [math.CO], 2019.

Programs

Maple

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

Mathematica

a[n_] := Length[IntegerPartitions[4*n, 8, Range[n]]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 17 2014 *)

Formula

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

Extensions

Edited by Henry Bottomley, Jul 19 2003