A001980 Number of partitions of floor(7n/2)-1 into n nonnegative integers each no greater than 7. (Formerly M3388 N1368)

0, 1, 4, 10, 23, 48, 94, 166, 285, 464, 734, 1109, 1646, 2371, 3366, 4652, 6357, 8519, 11309, 14754, 19103, 24399, 30956, 38797, 48355, 59665, 73264, 89145, 108011, 129864, 155554, 185017, 219336, 258438, 303604, 354665, 413213, 479048, 554033

Offset

0,3

Comments

In Cayley's terminology, this is the number of literal terms of degree n and of weight floor(7n/2)-1 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).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 86 terms from Vincenzo Librandi)

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, 1, -1, 2, -1, -2, 0, -1, 0, 1, 2, 2, -1, -2, 1, -4, 0, 4, -1, 2, 1, -2, -2, -1, 0, 1, 0, 2, 1, -2, 1, -1, -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)), where w=floor(7n/2)-1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: -(x^24 +3*x^23 +5*x^22 +10*x^21 +17*x^20 +26*x^19 +33*x^18 +45*x^17 +55*x^16 +61*x^15 +63*x^14 +68*x^13 +67*x^12 +68*x^11 +63*x^10 +61*x^9 +55*x^8 +45*x^7 +33*x^6 +26*x^5 +17*x^4 +10*x^3 +5*x^2 +3*x +1)*x / ((x^4+x^3+x^2+x+1) *(x^4-x^2+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

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)); n=400; 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)-1; print1(polcoeff(polcoeff(p, w), d)", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

Crossrefs

Cf. A001979.

Sequence in context: A084446 A209815 A158671 * A266376 A057750 A377823

Adjacent sequences: A001977 A001978 A001979 * A001981 A001982 A001983

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