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

Author

Jon Perry, Feb 06 2004

Keywords

• nonn

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

Examples

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

Crossrefs

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

Programs

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

Mathematica

nmax = 100; Rest[CoefficientList[Series[x + Sum[x^(4*k)*Product[(1 + x^(2*j - 1)), {j, 1, k-1}], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 15 2025 *)

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",")) }

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)

a(n) ~ Pi * exp(Pi*sqrt(n/6)) / (2^(9/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Jun 15 2025

Extensions

More terms from Vladeta Jovovic, Feb 10 2004