A001978 Number of partitions of 3n-1 into n nonnegative integers each no more than 6. (Formerly M2725 N1092)

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

0,3

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).

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

Prog

(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

Crossrefs

Cf. A001977.

Sequence in context: A188123 A081661 A005103 * A173283 A077552 A171497

Adjacent sequences: A001975 A001976 A001977 * A001979 A001980 A001981

Keyword

nonn,easy

Author

N. J. A. Sloane

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

Status

approved