A238212 The total number of 5's in all partitions of n into an odd number of distinct parts.

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 1, 2, 2, 3, 5, 4, 5, 7, 8, 10, 11, 13, 16, 19, 23, 26, 31, 36, 42, 49, 56, 65, 75, 86, 100, 114, 130, 149, 170, 193, 220, 250, 283, 321, 363, 410, 463, 522, 587, 660, 742, 832, 933, 1045, 1168, 1307, 1459, 1627, 1814, 2020

Offset

0,11

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/10)} A067661(n-(2*j-1)*5) - Sum_{j=1..floor(n/10)} A067659(n-10*j).

G.f.: (1/2)*(x^5/(1+x^5))*(Product_{n>=1} 1 + x^n) + (1/2)*(x^5/(1-x^5))*(Product_{n>=1} 1 - x^n).

Example

a(12) = 2 because the partitions in question are: 6+5+1, 5+4+3.

Mathematica

tn5[n_]:=Module[{op=IntegerPartitions[n], m}, m=Flatten[Select[op, OddQ[ Length[#]] && Length[#]==Length[Union[#]]&]]; Count[m, 5]]; Array[tn5, 60, 0] (* Harvey P. Dale, Feb 06 2015 *)

Crossrefs

Column k=5 of A238450.

Cf. A067659, A067661.

Sequence in context: A347662 A060177 A347788 * A255723 A309022 A214718

Adjacent sequences: A238209 A238210 A238211 * A238213 A238214 A238215

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

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

Status

approved