A238220 - OEIS

A238220

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

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 2, 2, 2, 3, 3, 5, 6, 7, 8, 9, 12, 14, 16, 19, 22, 27, 31, 36, 42, 48, 56, 65, 75, 86, 99, 114, 130, 149, 170, 193, 220, 250, 283, 321, 364, 410, 463, 522, 587, 661, 742, 832, 933, 1045, 1169, 1306, 1459, 1627, 1814, 2021

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

EXAMPLE

a(13) = 2 because the partitions in question are: 8+5, 5+4+3+1.

MATHEMATICA

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 *)

CROSSREFS

Column k=5 of A238451.

Cf. A067659, A067661.

Sequence in context: A367395 A065308 A035680 * A103600 A267160 A374436

Adjacent sequences: A238217 A238218 A238219 * A238221 A238222 A238223

KEYWORD

nonn

AUTHOR

Mircea Merca, Feb 20 2014

EXTENSIONS

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

STATUS

approved