A274325 Number of partitions of n^5 into at most two parts.

1, 1, 17, 122, 513, 1563, 3889, 8404, 16385, 29525, 50001, 80526, 124417, 185647, 268913, 379688, 524289, 709929, 944785, 1238050, 1600001, 2042051, 2576817, 3218172, 3981313, 4882813, 5940689, 7174454, 8605185, 10255575, 12150001, 14314576, 16777217

Offset: 0

Author

Colin Barker, Jun 18 2016

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Crossrefs

A subsequence of A008619.

Cf. A099392 (n^2), A274324 (n^3).

Programs

Magma

[(3+(-1)^n+2*n^5)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jun 25 2016

Maple

A274325:=n->(3+(-1)^n+2*n^5)/4: seq(A274325(n), n=0..50); # Wesley Ivan Hurt, Jun 25 2016

Mathematica

Table[(3+(-1)^n+2*n^5)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 25 2016 *)

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^5))

Formula

Coefficient of x^(n^5) in 1/((1-x)*(1-x^2)).

a(n) = A008619(n^5).

a(n) = (3 + (-1)^n + 2*n^5)/4.

a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7) for n > 6.

G.f.: (1 - 4*x + 21*x^2 + 41*x^3 + 46*x^4 + 15*x^5) / ((1-x)^6*(1+x)).

E.g.f.: ((2 + x + 15*x^2 + 25*x^3 + 10*x^4 + x^5)*cosh(x) + (1 + x + 15*x^2 + 25*x^3 + 10*x^4 + x^5)*sinh(x))/2. - Stefano Spezia, Mar 17 2024