A001975 Number of partitions of floor(5n/2) into n nonnegative integers each no more than 5.

1, 1, 3, 6, 12, 20, 32, 49, 73, 102, 141, 190, 252, 325, 414, 521, 649, 795, 967, 1165, 1394, 1651, 1944, 2275, 2649, 3061, 3523, 4035, 4604, 5225, 5910, 6660, 7483, 8372, 9343, 10395, 11538, 12764, 14090, 15516, 17053, 18691, 20451, 22330, 24342, 26476, 28754

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 floor(5n/2) involving the letters a, b, c, d, e, f, having weights 0, 1, 2, 3, 4, 5 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]
• 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,-2,0,2,-2,2,-2,2,-1,0,1,-2,1).

Programs

Mathematica

LinearRecurrence[{2, -1, 0, 1, -2, 2, -2, 2, -2, 0, 2, -2, 2, -2, 2, -1, 0, 1, -2, 1}, {1, 1, 3, 6, 12, 20, 32, 49, 73, 102, 141, 190, 252, 325, 414, 521, 649, 795, 967, 1165}, 50] (* Jean-François Alcover, Feb 26 2020 *)

PARI

f=1/((1-z)*(1-x*z)*(1-x^2*z)*(1-x^3*z)*(1-x^4*z)*(1-x^5*z)); n=350; p=subst(subst(f,x,x+x*O(x^n)),z,z+z*O(z^n)); for(d=0,60,w=floor(5*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)), where w=floor(5n/2). - Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 17 2008

G.f.: -(x^14 -x^13 +2*x^12 +x^11 +2*x^10 +3*x^9 +x^8 +5*x^7 +x^6 +3*x^5 +2*x^4 +x^3 +2*x^2 -x+1) / ((x^4+1) *(x^2+x+1) *(x^2-x+1) *(x^2+1)^2 *(x+1)^3 *(x-1)^5). - Alois P. Heinz, Jul 25 2015

Extensions

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