A001978 Number of partitions of 3n-1 into n nonnegative integers each no more than 6.

0, 1, 3, 8, 16, 32, 55, 94, 147, 227, 332, 480, 668, 920, 1232, 1635, 2124, 2738, 3470, 4368, 5424, 6695, 8172, 9922, 11934, 14287, 16968, 20068, 23572, 27584, 32087, 37199, 42901, 49325, 56450, 64424, 73223, 83012, 93764, 105661, 118674, 133003, 148616

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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]
• Index entries for linear recurrences with constant coefficients, signature (1, 3, -2, -4, 1, 3, -1, -1, 3, 1, -4, -2, 3, 1, -1).

Crossrefs

Cf. A001977.

Programs

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)); n=400; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=3*d-1;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)), where w=3n-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: (x^6 +2*x^5 +2*x^4 +x^3 +2*x^2 +2*x+1)*x / ((x^2+x+1) *(x^4+x^3+x^2+x+1) *(x+1)^3 *(x-1)^6). - Alois P. Heinz, Jul 25 2015

Extensions

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

a(0)=0 inserted by Alois P. Heinz, Jul 25 2015