A014126 Number of partitions of 2*n into at most 4 parts. (Formerly N0523)

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

0,2

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).

Formula

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A308265 A218914 A047809 * A019450 A098169 A055610

Adjacent sequences: A014123 A014124 A014125 * A014127 A014128 A014129

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved