A025157 Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 3.

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 22, 25, 28, 32, 36, 41, 46, 52, 58, 66, 73, 82, 91, 102, 113, 126, 139, 155, 171, 190, 209, 232, 255, 282, 310, 342, 375, 413, 452, 497, 544, 596, 651, 713, 778, 850, 927, 1011, 1101, 1200, 1305, 1420, 1544, 1677, 1821, 1977, 2144, 2324, 2519, 2728

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Also number of partitions of n into distinct parts in which the smallest part is greater than or equal to number of parts. - Vladeta Jovovic, Aug 06 2004

Examples

			a(12) = 6 because we have 12 = 11+1 = 10+2 = 9+3 = 8+4 = 7+4+1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• I. Martinjak and D. Svrtan, New Identities for the Polarized Partitions and Partitions with d-Distant Parts, J. Int. Seq. 17 (2014) # 14.11.4.

Crossrefs

Cf. A003114, A096401, A263719.

Column k=3 of A194543.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(n>
      ceil(i*(i+3)/6), 0, b(n, i-1)+b(n-i, min(n-i, i-3))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 26 2022

Mathematica

nn=50; CoefficientList[Series[Sum[x^(j(3j-1)/2)Product[1/(1-x^i), {i, 1, j}], {j, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Jul 21 2013 *)

PARI

my(N=99, x='x+O('x^N)); Vec(sum(k=0, N, x^(k*(3*k-1)/2)/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 13 2022

Formula

G.f.: sum(i>=1, x^(3*A000217(i)-2*i)/product(j=1..i, 1-x^j)). - Jon Perry, Jul 20 2004

G.f.: sum(n>=0, x^(n*(3*n-1)/2)/prod(k=1..n,1-x^k)). - Joerg Arndt, Jan 29 2011

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*n^(3/4)*r*sqrt(Pi*(1+3*r^2))), where r = A263719 = ((9+sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9+sqrt(93))))^(1/3) = 0.682327803828019327369483739711048256891188581898... is the root of the equation r^3 + r = 1 and c = 3*(log(r))^2/2 + polylog(2, 1-r) = 0.566433354765746647188050807325058683443823543741343518... . - Vaclav Kotesovec, Jan 02 2016

Extensions

Prepended a(0)=1, Joerg Arndt, Jul 21 2013