A001979 Number of partitions of floor(7n/2) into n nonnegative integers each no more than 7.

1, 1, 4, 10, 24, 49, 94, 169, 289, 468, 734, 1117, 1656, 2385, 3370, 4672, 6375, 8550, 11322, 14800, 19138, 24460, 30982, 38882, 48417, 59779, 73316, 89291, 108108, 130053, 155646, 185258, 219489, 258735, 303748, 355034, 413442, 479500, 554256

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• 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.
• 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.
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Cf. A001980.

Programs

Maple

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

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

Formula

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

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

Extensions

Better definition and more terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008