A394985 Number of 1's in the partitions of n into exactly 6 parts.

0, 0, 0, 0, 0, 0, 6, 5, 9, 12, 18, 22, 31, 37, 49, 59, 74, 87, 108, 125, 150, 174, 205, 234, 273, 309, 356, 401, 456, 510, 577, 641, 718, 795, 885, 974, 1079, 1182, 1302, 1422, 1558, 1695, 1851, 2006, 2181, 2358, 2555, 2753, 2974, 3196, 3442, 3691, 3963, 4239, 4542, 4847, 5180

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

Formula

G.f.: q^6 * Sum_{j=0..5} 1 / Product_{k=1..j} (1 - q^k).

G.f.: q^6 * Sum_{j=0..5} (6-j) * q^j / Product_{k=1..j} (1 - q^k).

a(n) = a(n-1) + a(n-2) - a(n-5) - a(n-6) - a(n-7) + a(n-8) + a(n-9) + a(n-10) - a(n-13) - a(n-14) + a(n-15) for n > 21.

Example

The partitions of 8 into exactly 6 parts are:
311111 contains five 1's.
221111 contains four 1's.
So a(8) = 5 + 4 = 9.

Prog

(PARI) my(N=60, q='q+O('q^N)); concat([0, 0, 0, 0, 0, 0], Vec(q^6*sum(j=0, 5, 1/prod(k=1, j, 1-q^k))))

Crossrefs

Column k=6 of A394828.

Cf. A026812.

Sequence in context: A134881 A229983 A390523 * A251859 A100884 A309549

Adjacent sequences: A394982 A394983 A394984 * A394986 A394987 A394988

Keyword

nonn,easy

Author

Seiichi Manyama, May 13 2026

Status

approved