A238219 The total number of 4's in all partitions of n into an even number of distinct parts.

0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 2, 3, 4, 4, 5, 6, 8, 9, 11, 13, 16, 18, 21, 25, 29, 34, 40, 46, 53, 62, 71, 82, 94, 108, 124, 142, 161, 185, 210, 238, 270, 307, 347, 392, 442, 499, 562, 632, 709, 797, 894, 1000, 1119, 1252, 1398, 1560, 1739, 1937, 2157

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/8)} A067659(n-(2*j-1)*4) - Sum_{j=1..floor(n/8)} A067661(n-8*j).

G.f.: (1/2)*(x^4/(1+x^4))*(Product_{n>=1} 1 + x^n) - (1/2)*(x^4/(1-x^4))*(Product_{n>=1} 1 - x^n).

Example

a(13) = 3 because the partitions in question are: 9+4, 6+4+2+1, 5+4+3+1.

Crossrefs

Column k=4 of A238451.

Cf. A067659, A067661.

Sequence in context: A030383 A031231 A030562 * A026833 A281544 A056882

Adjacent sequences: A238216 A238217 A238218 * A238220 A238221 A238222

Keyword

nonn

Author

Mircea Merca, Feb 20 2014

Extensions

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

Status

approved