A271638 The total sum of the cubes of all parts of all compositions of n.

1, 10, 48, 170, 512, 1398, 3580, 8770, 20808, 48206, 109652, 245850, 544864, 1196134, 2605164, 5636210, 12124280, 25952382, 55312516, 117440650, 248512656, 524288150, 1103102108, 2315255970, 4848615592, 10133438638, 21139292340, 44023414970, 91536490688

Offset

1,2

Links

Table of n, a(n) for n=1..29.

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Formula

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

a(n) = (13*n - 36)*2^(n - 1) + 6*n + 18.

Example

The two compositions of n=2 are 2 and 1+1. The total sum of the cubes is a(2) = 2^3+1^3+1^3 = 10.

Mathematica

Table[(13 n - 36) 2^(n - 1) + 6 n + 18, {n, 29}] (* or *)
Rest@ CoefficientList[Series[x (1 + 4 x + x^2)/((1 - 2 x) (1 - x))^2, {x, 0, 29}], x] (* Michael De Vlieger, Apr 11 2016 *)

Prog

(PARI) x='x+O('x^99); Vec(x*(1+4*x+x^2)/((2*x-1)*(1-x))^2) \\ Altug Alkan, Apr 11 2016
(Python) for n in range(1, 50):print((13*n-36)*2**(n-1)+6*n+18) # Soumil Mandal, Apr 11 2016

Crossrefs

Cf. A027992 (sum of squares).

Sequence in context: A353620 A277229 A163724 * A238916 A084857 A349948

Adjacent sequences: A271635 A271636 A271637 * A271639 A271640 A271641

Keyword

nonn,easy

Author

R. J. Mathar, Apr 11 2016

Status

approved