A396011 Number of 2's in the partitions of n into exactly 5 parts.

0, 0, 0, 0, 0, 0, 1, 2, 4, 7, 12, 13, 18, 22, 28, 33, 41, 47, 57, 65, 76, 86, 100, 111, 127, 141, 159, 175, 196, 214, 238, 259, 285, 309, 339, 365, 398, 428, 464, 497, 537, 573, 617, 657, 704, 748, 800, 847, 903, 955, 1015, 1071, 1136, 1196, 1266, 1331, 1405, 1475, 1555

Offset

0,8

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

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^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 > 20.

a(n) = A320592(n+2) - 1 for n > 10.

Mathematica

m=60; a=CoefficientList[Series[x^5 Sum[x^j/Product[1-x^k, {k, 1, 5-j}], {j, 1, 5}], {x, 0, m}], x]; a (* Vincenzo Librandi, May 22 2026 *)

Prog

(PARI) my(N=60, q='q+O('q^N)); concat([0, 0, 0, 0, 0, 0], Vec(q^5*sum(j=1, 5, q^j/prod(k=1, 5-j, 1-q^k))))
(Magma) N := 60; R<q> := PowerSeriesRing(Integers(), N); a := [0, 0, 0, 0, 0, 0] cat Coefficients( q^5 * &+[ q^j / (5-j eq 0 select 1 else &*[1-q^k : k in [1..5-j]]) : j in [1..5] ] ); a; // Vincenzo Librandi, May 22 2026

Crossrefs

Cf. A394873, A396012.

Cf. A024786, A026811, A320592, A394827.

Sequence in context: A394573 A052474 A033054 * A361724 A359338 A382832

Adjacent sequences: A396008 A396009 A396010 * A396012 A396013 A396014

Keyword

nonn,easy

Author

Seiichi Manyama, May 13 2026

Status

approved