A003404 Number of solid partitions of n supported on graph of cube.

1, 1, 4, 7, 14, 23, 41, 63, 104, 152, 230, 327, 470, 647, 897, 1202, 1616, 2117, 2775, 3566, 4580, 5787, 7301, 9092, 11298, 13885, 17028, 20688, 25076, 30154, 36172, 43094, 51221, 60511, 71323, 83622, 97822, 113893, 132323, 153083

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

References

• P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373 (see Section 98).
• J. C. P. Miller, On the enumeration of partially ordered sets of integers, pp. 109-124 of T. P. McDonough and V. C. Mavron, editors, Combinatorics: Proceedings of the Fourth British Combinatorial Conference 1973. London Mathematical Society, Lecture Note Series, Number 13, Cambridge University Press, NY, 1974. [The g.f. shown below appears on page 121. - N. J. A. Sloane, Apr 22 2015]
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Matthew House, Table of n, a(n) for n = 0..10000
• G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 14.
• G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis: The Omega Package, Europ. J. Combin., 22 (2001), 887-904.
• Index entries for sequences related to posets
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-1,0,-1,0,-1,0,1,2,1,0,1,-1,-1,-2,-1,-1,1,0,1,2,1,0,-1,0,-1,0,-1,0,0,1,1,-1).

Crossrefs

Cf. A003402, A003403, A003405, A029073, A256975, A256976, A256977.

Programs

Mathematica

CoefficientList[Series[(1+2*q^2+2*q^3+3*q^4+3*q^5+5*q^6+4*q^7+8*q^8+ 4*q^9+ 5*q^10+ 3*q^11+3*q^12+2*q^13+2*q^14+q^16)/((1-q)*(1-q^2)*(1-q^3)*(1-q^4)* (1-q^5)*(1-q^6)*(1-q^7)*(1-q^8)),{q,0,40}],q] (* Harvey P. Dale, Mar 07 2012 *)
LinearRecurrence[{1,1,0,0,-1,0,-1,0,-1,0,1,2,1,0,1,-1,-1,-2,-1,-1,1,0,1,2,1,0,-1,0,-1,0,-1,0,0,1,1,-1},{1,1,4,7,14,23,41,63,104,152,230,327,470,647,897,1202,1616,2117,2775,3566,4580,5787,7301,9092,11298,13885,17028,20688,25076,30154,36172,43094,51221,60511,71323,83622},50] (* Harvey P. Dale, Jun 11 2022 *)

Formula

G.f.: (1 + 2*q^2 + 2*q^3 + 3*q^4 + 3*q^5 + 5*q^6 + 4*q^7 + 8*q^8 + 4*q^9 + 5*q^10 + 3*q^11 + 3*q^12 + 2*q^13 + 2*q^14 + q^16)/((1 - q)*(1 - q^2)*(1 - q^3)*(1 - q^4)*(1 - q^5)*(1 - q^6)*(1 - q^7)*(1 - q^8)).