A238221 The total number of 6's in all partitions of n into an even number of distinct parts.

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 3, 3, 4, 4, 6, 7, 9, 11, 12, 14, 17, 20, 24, 28, 32, 37, 44, 51, 59, 69, 78, 90, 104, 119, 136, 156, 177, 202, 230, 261, 296, 336, 379, 428, 483, 544, 612, 689, 773, 867, 972, 1088, 1217, 1360, 1518, 1693, 1887

Offset

0,14

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

Example

a(13) = 2 because the partitions in question are: 7+6, 6+4+2+1.

Mathematica

endpQ[n_]:=Module[{len=Length[n]}, EvenQ[len]&&len==Length[Union[n]]]; Table[ Count[Flatten[Select[IntegerPartitions[i], endpQ]], 6], {i, 0, 50}] (* Harvey P. Dale, Mar 03 2014 *)

Crossrefs

Column k=6 of A238451.

Cf. A067659, A067661.

Sequence in context: A103297 A274017 A267597 * A320052 A168173 A095916

Adjacent sequences: A238218 A238219 A238220 * A238222 A238223 A238224

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

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

Status

approved