A396009 Number of 2's in the partitions of n into exactly 4 parts.

0, 0, 0, 0, 0, 1, 2, 4, 7, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 32, 36, 39, 43, 47, 51, 55, 60, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 119, 126, 132, 139, 146, 153, 160, 168, 175, 183, 191, 199, 207, 216, 224, 233, 242, 251, 260, 270, 279, 289, 299, 309, 319, 330

Offset

0,7

Links

Seiichi Manyama, 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,-1,-1,1).

Formula

G.f.: q^4 * Sum_{j=1..4} q^j / Product_{k=1..4-j} (1 - q^k).

a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 14.

a(n) = A160138(n-1) for n > 8.

Mathematica

m=62; CoefficientList[Series[x^4 Sum[x^j/Product[1-x^k, {k, 1, 4-j}], {j, 1, 4}], {x, 0, m}], x] (* Vincenzo Librandi, May 23 2026 *)

Prog

(PARI) my(N=70, q='q+O('q^N)); concat([0, 0, 0, 0, 0], Vec(q^4*sum(j=1, 4, q^j/prod(k=1, 4-j, 1-q^k))))
(Magma) N := 63; R<q> := PowerSeriesRing(Integers(), N); a := [0, 0, 0, 0, 0] cat Coefficients( q^4 * &+[ q^j / (4-j eq 0 select 1 else &*[1-q^k : k in [1..4-j]]) : j in [1..4] ] ); a; // Vincenzo Librandi, May 23 2026

Crossrefs

Cf. A335583, A396010.

Cf. A024786, A160138, A244239, A253186.

Sequence in context: A363166 A393402 A161368 * A023978 A152488 A109359

Adjacent sequences: A396006 A396007 A396008 * A396010 A396011 A396012

Keyword

nonn,easy,changed

Author

Seiichi Manyama, May 13 2026

Status

approved