A274324 Number of partitions of n^3 into at most two parts.

1, 1, 5, 14, 33, 63, 109, 172, 257, 365, 501, 666, 865, 1099, 1373, 1688, 2049, 2457, 2917, 3430, 4001, 4631, 5325, 6084, 6913, 7813, 8789, 9842, 10977, 12195, 13501, 14896, 16385, 17969, 19653, 21438, 23329, 25327, 27437, 29660, 32001, 34461, 37045, 39754

Offset

0,3

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

Coefficient of x^(n^3) in 1/((1-x)*(1-x^2)).

a(n) = A008619(n^3).

a(n) = (3+(-1)^n+2*n^3)/4.

a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4.

G.f.: (1-2*x+4*x^2+3*x^3) / ((1-x)^4*(1+x)).

From Stefano Spezia, Sep 28 2022: (Start)

a(n) = A050492((n+1)/2) for n odd.

E.g.f.: ((2 + x + 3*x^2 + x^3)*cosh(x) + (1 + x + 3*x^2 + x^3)*sinh(x))/2. (End)

Maple

A274324:=n->(3+(-1)^n+2*n^3)/4: seq(A274324(n), n=0..50); # Wesley Ivan Hurt, Jun 25 2016

Mathematica

Table[(3+(-1)^n+2*n^3)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 25 2016 *)

Prog

(PARI)
\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)).
b(n) = (3+(-1)^n+2*n)/4
vector(50, n, n--; b(n^3))
(Magma) [(3+(-1)^n+2*n^3)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jun 25 2016

Crossrefs

A subsequence of A008619.

Cf. A099392 (n^2), A274325 (n^5).

Cf. also A050492.

Sequence in context: A053209 A306192 A271993 * A014302 A038090 A094002

Adjacent sequences: A274321 A274322 A274323 * A274325 A274326 A274327

Keyword

nonn,easy

Author

Colin Barker, Jun 18 2016

Status

approved