A001982 Number of partitions of 4n-1 into n nonnegative integers each no greater than 8. (Formerly M3441 N1396)

0, 1, 4, 12, 31, 71, 147, 285, 519, 902, 1502, 2417, 3768, 5722, 8481, 12310, 17528, 24537, 33814, 45949, 61629, 81688, 107089, 138979, 178669, 227703, 287828, 361075, 449731, 556423, 684089, 836078, 1016110, 1228391, 1477573, 1768875, 2108041, 2501480

Offset

0,3

Comments

In Cayley's terminology, this is the number of literal terms of degree n and of weight 4n-1 involving the letters a, b, c, d, e, f, g, h, i, having weights 0, 1, 2, 3, 4, 5, 6, 7, 8 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, -1, -5, -2, 5, 4, -3, -3, 3, 2, -3, -3, 2, 3, -3, -3, 4, 5, -2, -5, -1, 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)(1-x^7z)(1-x^8z)), where w=4n-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: (x^14 +3*x^13 +5*x^12 +8*x^11 +13*x^10 +17*x^9 +19*x^8 +19*x^7 +19*x^6 +17*x^5 +13*x^4 +8*x^3 +5*x^2 +3*x+1)*x / ((x^4+x^3+x^2+x+1) *(x^6+x^5+x^4+x^3+x^2+x+1) *(x^2+x+1)^2 *(x+1)^3 *(x-1)^8). - 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)*(1-x^7*z)*(1-x^8*z)); n=400; p=subst(subst(f, x, x+x*O(x^n)), z, z+z*O(z^n)); for(d=0, 60, w=4*d-1; print1(polcoeff(polcoeff(p, w), d)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

Crossrefs

Cf. A001981.

Sequence in context: A005289 A037255 A027658 * A129707 A320545 A232580

Adjacent sequences: A001979 A001980 A001981 * A001983 A001984 A001985

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