A396012 Number of 3's in the partitions of n into exactly 5 parts.

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 16, 19, 27, 30, 38, 44, 54, 61, 73, 82, 96, 107, 123, 136, 155, 170, 191, 209, 233, 253, 280, 303, 333, 359, 392, 421, 458, 490, 530, 566, 610, 649, 697, 740, 792, 839, 895, 946, 1007, 1062, 1127, 1187, 1257, 1321, 1396, 1465, 1545, 1619, 1704, 1783

Offset

0,10

Links

Table of n, a(n) for n=0..62.

Index entries for sequences related to partitions

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

Formula

G.f.: q^5 * Sum_{j=1..5} q^(2*j) / Product_{k=1..5-j} (1 - q^k).

a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10) for n > 25.

Prog

(PARI) my(N=70, q='q+O('q^N)); concat([0, 0, 0, 0, 0, 0, 0], Vec(q^5*sum(j=1, 5, q^(2*j)/prod(k=1, 5-j, 1-q^k))))

Crossrefs

Cf. A394873, A396011.

Cf. A026811.

Sequence in context: A026494 A043306 A308844 * A131355 A092534 A005232

Adjacent sequences: A396009 A396010 A396011 * A396013 A396014 A396015

Keyword

nonn,easy

Author

Seiichi Manyama, May 13 2026

Status

approved