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

Author

Mircea Merca, Feb 20 2014

Keywords

• nonn

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).

Examples

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

Links

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

Crossrefs

Column k=5 of A238450.

Cf. A067659, A067661.

Programs

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 *)
nmax = 100; With[{k=5}, 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 *)

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).

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

Extensions

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