A238213 The total number of 6's in all partitions of n into an odd number of distinct parts.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 2, 2, 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, 17, 20, 23, 27, 33, 38, 44, 51, 59, 68, 79, 91, 104, 119, 136, 155, 178, 202, 230, 261, 296, 335, 379, 428, 483, 544, 612, 688, 773, 867, 972, 1088, 1217, 1360, 1518, 1693, 1887

Offset

0,12

Comments

The g.f. for "number of k's" is (1/2)*(x^k/(1+x^k))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^k/(1-x^k))*(Product_{n>=1} 1 - x^n).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{j=1..round(n/12)} A067661(n-(2*j-1)*6) - Sum_{j=1..floor(n/12)} A067659(n-12*j).

G.f.: (1/2)*(x^6/(1+x^6))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^6/(1-x^6))*(Product_{n>=1} 1 - x^n).

a(n) ~ exp(Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jul 05 2025

Example

a(12) = 2 because the partitions in question are: 6+5+1, 6+4+2.

Mathematica

nmax = 100; With[{k=6}, CoefficientList[Series[x^k/(1+x^k)/2 * Product[1 + x^j, {j, 1, nmax}] + x^k/(1-x^k)/2 * Product[1 - x^j, {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jul 05 2025 *)

Crossrefs

Column k=6 of A238450.

Cf. A067659, A067661.

Sequence in context: A147652 A058360 A241901 * A193942 A098527 A035635

Adjacent sequences: A238210 A238211 A238212 * A238214 A238215 A238216

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

Terms a(51) and beyond from Andrew Howroyd, Apr 29 2020

Status

approved