A090723 a(1) = 1; for n > 1, number of partitions of n into distinct odd parts such that the two largest parts differ by 2.

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 1, 3, 3, 1, 2, 4, 3, 2, 3, 5, 5, 3, 4, 7, 6, 4, 6, 9, 8, 6, 8, 11, 11, 9, 10, 15, 15, 11, 14, 19, 18, 16, 19, 24, 24, 21, 24, 31, 31, 27, 32, 40, 39, 36, 41, 49, 50, 47, 52, 63, 64, 59, 67, 79, 79, 77, 85, 98, 101, 97, 106

Offset

1,16

Comments

Also number of partitions of n such that if k is the largest part, then 1 occurs twice, each of 2,3,...,k-1 occurs a positive even number of times and k occurs an odd number of times. Example: a(25)=3 because we have [5,4,4,3,3,2,2,1,1], [3,3,3,3,3,2,2,2,2,1,1] and [3,2,2,2,2,2,2,2,2,2,2,1,1]. - Emeric Deutsch, Mar 29 2006

Links

Table of n, a(n) for n=1..83.

Formula

From Emeric Deutsch, Mar 29 2006: (Start)

G.f.: x+sum(x^(4k)*product(1+x^(2j-1), j=1..k-1), k=1..infinity).

G.f.: sum(x^(k^2)/product(1-x^(2j), j=2..k), k=1..infinity). (End)

Example

a(25)=3 as 25=1+3+5+7+9=5+9+11=1+11+13.

Maple

g:=sum(x^(k^2)/product(1-x^(2*j), j=2..k), k=1..10): gser:=series(g, x=0, 95): seq(coeff(gser, x, n), n=1..92); # Emeric Deutsch, Mar 29 2006

Prog

(PARI) { v=[1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 1, 3, 3, 1, 2, 4, 3, 2]; for (i=2, 30, c=0; forstep (j=i, 1, -2, c+=v[j]); print1(c", ")) }

Crossrefs

a(n) = A000700(n) - A000700(n-2), n > 2.

Sequence in context: A282318 A286971 A025861 * A027357 A256479 A277078

Adjacent sequences: A090720 A090721 A090722 * A090724 A090725 A090726

Keyword

nonn

Author

Jon Perry, Feb 06 2004

Extensions

More terms from Vladeta Jovovic, Feb 10 2004

Status

approved