A288166 Expansion of x^5/((1-x^5)*(1-x^4)*(1-x^8)*(1-x^12)*(1-x^16)).

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 3, 2, 1, 1, 5, 3, 2, 1, 7, 5, 3, 2, 10, 7, 5, 3, 13, 10, 7, 5, 18, 13, 10, 7, 23, 18, 13, 10, 30, 23, 18, 13, 37, 30, 23, 18, 47, 37, 30, 23, 57, 47, 37, 30, 70, 57, 47, 37, 84, 70, 57, 47, 101, 84, 70, 57, 119

Offset

0,14

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Daniel Panario, Murat Sahin and Qiang Wang, Generalized Alcuin’s Sequence, The Electronic Journal of Combinatorics, Volume 19, Issue 4 (2012).

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 1, 1, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, -1, -1, 0, 0, 0, 1).

Formula

a(n) = p_5(n/4) if n == 0 mod 4,

a(n) = p_5((n+15)/4) if n == 1 mod 4,

a(n) = p_5((n+10)/4) if n == 2 mod 4,

a(n) = p_5((n+5)/4) if n == 3 mod 4,

where p_5(n) is the number of partitions of n into exactly 5 parts.

Example

a(56) = p_5(56/4)      = p_5(14) = A001401(9)  = 23,
a(57) = p_5((57+15)/4) = p_5(18) = A001401(13) = 57,
a(58) = p_5((58+10)/4) = p_5(17) = A001401(12) = 47,
a(59) = p_5((59+5)/4)  = p_5(16) = A001401(11) = 37,
a(60) = p_5(60/4)      = p_5(15) = A001401(10) = 30,
a(61) = p_5((61+15)/4) = p_5(19) = A001401(14) = 70,
a(62) = p_5((62+10)/4) = p_5(18) = A001401(13) = 57,
a(63) = p_5((63+5)/4)  = p_5(17) = A001401(12) = 47.

Mathematica

CoefficientList[Series[x^5/((1-x^4)(1-x^5)(1-x^8)(1-x^12)(1-x^16)), {x, 0, 120}], x] (* or *) LinearRecurrence[ {0, 0, 0, 1, 1, 0, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, -1, -1, 0, 0, 0, 1}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 3, 2, 1, 1, 5, 3, 2, 1, 7, 5, 3, 2, 10, 7, 5, 3, 13, 10, 7, 5, 18, 13, 10, 7, 23, 18, 13, 10}, 120] (* Harvey P. Dale, Apr 22 2019 *)

Crossrefs

Cf. A001401.

Cf. A005044 (k=3), A288165 (k=4), this sequence (k=5).

Sequence in context: A279209 A087117 A029340 * A126258 A235501 A116382

Adjacent sequences: A288163 A288164 A288165 * A288167 A288168 A288169

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 06 2017

Status

approved