A003404 Number of solid partitions of n supported on graph of cube. (Formerly M3310)

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

0,3

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).

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)).

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 *)

Crossrefs

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

Sequence in context: A008370 A291877 A048241 * A139025 A221107 A128610

Adjacent sequences: A003401 A003402 A003403 * A003405 A003406 A003407

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Status

approved