A274323 Number of partitions of n^4 into at most two parts.

1, 1, 9, 41, 129, 313, 649, 1201, 2049, 3281, 5001, 7321, 10369, 14281, 19209, 25313, 32769, 41761, 52489, 65161, 80001, 97241, 117129, 139921, 165889, 195313, 228489, 265721, 307329, 353641, 405001, 461761, 524289, 592961, 668169, 750313, 839809, 937081

Offset: 0

Author

Colin Barker, Oct 13 2016

Keywords

• nonn
• easy

Comments

Coefficient of x^(n^4) in 1/((1-x)*(1-x^2)).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Crossrefs

Cf. A099392 (n^2), A274324 (n^3), A274325 (n^5).

Cf. A008619.

Programs

PARI

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

PARI

b(n) = (3+(-1)^n+2*n)/4 \\ the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2))
vector(50, n, n--; b(n^4))

Formula

G.f.: (1 - 3*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5) / ((1-x)^5*(1+x)).

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

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

a(n) = A008619(n^4).

a(n) = 1 + floor(n^4/2). - Alois P. Heinz, Oct 13 2016

E.g.f.: ((2 + x + 7*x^2 + 6*x^3 + x^4)*cosh(x) + (1 + x + 7*x^2 + 6*x^3 + x^4)*sinh(x))/2. - Stefano Spezia, Mar 17 2024