A288165 Expansion of x^4/((1-x^4)*(1-x^3)*(1-x^6)*(1-x^9)).

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1, 1, 3, 2, 1, 5, 3, 2, 6, 5, 3, 9, 6, 5, 11, 9, 6, 15, 11, 9, 18, 15, 11, 23, 18, 15, 27, 23, 18, 34, 27, 23, 39, 34, 27, 47, 39, 34, 54, 47, 39, 64, 54, 47, 72, 64, 54, 84, 72, 64, 94, 84, 72, 108, 94, 84, 120, 108, 94, 136, 120

Offset: 0

Author

Seiichi Manyama, Jun 06 2017

Keywords

• nonn
• easy

Examples

			a(57) = p_4(57/3)     = p_4(19) = A001400(15) = 54,
a(58) = p_4((58+8)/3) = p_4(22) = A001400(18) = 84,
a(59) = p_4((59+4)/3) = p_4(21) = A001400(17) = 72,
a(60) = p_4(60/3)     = p_4(20) = A001400(16) = 64,
a(61) = p_4((61+8)/3) = p_4(23) = A001400(19) = 94,
a(62) = p_4((62+4)/3) = p_4(22) = A001400(18) = 84.
		

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, 1, 1, 0, 1, -1, 0, 0, -1, 0, -1, 0, 0, -1, 1, 0, 1, 1, 0, 0, -1).

Crossrefs

Cf. A001400, A029253.

Cf. A005044 (k=3), this sequence (k=4), A288166 (k=5).

Formula

a(n) = p_4(n/3) if n == 0 mod 3,

a(n) = p_4((n+8)/3) if n == 1 mod 3,

a(n) = p_4((n+4)/3) if n == 2 mod 3,

where p_4(n) is the number of partitions of n into exactly 4 parts.