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

0,3

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).

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

Prog

(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))

Crossrefs

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

Cf. A008619.

Sequence in context: A245932 A373517 A362293 * A297740 A297741 A001846

Adjacent sequences: A274320 A274321 A274322 * A274324 A274325 A274326

Keyword

nonn,easy

Author

Colin Barker, Oct 13 2016

Status

approved