A025158 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 4.

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 11, 12, 14, 16, 18, 20, 23, 25, 28, 31, 35, 38, 43, 47, 53, 58, 65, 71, 80, 87, 97, 106, 118, 128, 142, 154, 170, 185, 203, 220, 242, 262, 287, 311, 340, 368, 402, 435, 474, 513, 558, 603, 656, 708, 768, 829, 898, 968, 1048

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

Also number of partitions of n such that if k is the largest part, then each 1,2,...,k-1 occur at least 4 times. Example: a(8)=3 because we have [2,2,1,1,1,1], [2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 17 2006

Examples

			a(8) = 3 because we have [8], [7,1] and [6,2].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A003114, A025157-A025162.

Column k=4 of A194543.

Programs

Maple

g:=sum(x^(2*k^2-k)/product(1-x^j,j=1..k),k=1..7): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=1..66); # Emeric Deutsch, Apr 17 2006

Mathematica

nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(j*(2*j - 1))/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 02 2016 *)

Formula

G.f.: Sum(x^(2*k^2-k)/Product(1-x^i, i=1..k), k=1..infinity). - Vladeta Jovovic, Aug 12 2004

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*sqrt(Pi*r^3*(1+4*r^3))), where r = 0.72449195900051561158837228218703656578649448135001101727... is the root of the equation r^4 + r = 1 and c = 2*log(r)^2 + polylog(2, 1-r) = 0.50498141294472195442598916817438524920370382784609501495065... . - Vaclav Kotesovec, Jan 02 2016

Extensions

More terms from Vladeta Jovovic, Aug 12 2004