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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
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
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
KEYWORD
nonn
AUTHOR
Jon Perry, Feb 06 2004
EXTENSIONS
More terms from Vladeta Jovovic, Feb 10 2004
STATUS
approved