A014126 Number of partitions of 2*n into at most 4 parts.

1, 2, 5, 9, 15, 23, 34, 47, 64, 84, 108, 136, 169, 206, 249, 297, 351, 411, 478, 551, 632, 720, 816, 920, 1033, 1154, 1285, 1425, 1575, 1735, 1906, 2087, 2280, 2484, 2700, 2928, 3169, 3422, 3689, 3969, 4263, 4571, 4894, 5231, 5584, 5952, 6336, 6736, 7153, 7586

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Bisection of A001400.

Molien series for 4-dimensional group of structure S_4 X C_2 and order 48, arising from complete weight enumerators of even trace-Hermitian self-dual additive codes over GF(4) containing the all-ones vector.

Partial sums of A156040. - Bob Selcoe, Feb 08 2014

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
• H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
• H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-685. (Annotated scanned copy)
• G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
• Index entries for Molien series
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1).

Crossrefs

Cf. A000631, A001400, A014125, A049347, A092498, A156040.

Programs

Maple

with(combstruct): seq(count(Partition((2*n+4)), size=4), n=0..50); # Zerinvary Lajos, Mar 28 2008

Mathematica

CoefficientList[Series[(1 + x^2) / ((1 - x)^2 (1 - x^2) (1 - x^3)), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 15 2013 *)
LinearRecurrence[{2,0,-1,-1,0,2,-1},{1,2,5,9,15,23,34},50] (* Harvey P. Dale, Aug 31 2015 *)

PARI

a(n)=(4*n^3+30*n^2+72*n+55+8*[1,-1,0][(n%3)+1]+9*(-1)^n)/72

Formula

G.f.: (1+x^2)/((1-x)^2*(1-x^2)*(1-x^3)). - James Sellers

a(n) = (1/72) * (4*n^3 + 30*n^2 + 72*n + 55 + 8*A049347(n) + 9*(-1)^n ). - Ralf Stephan, Aug 15 2013

E.g.f.: exp(-x)*(27 + 3*exp(2*x)*(55 + 106*x + 42*x^2 + 4*x^3) + 8*exp(x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/216. - Stefano Spezia, Apr 05 2023