A128508 Number of partitions p of n such that max(p) - min(p) = 3.

0, 0, 0, 0, 0, 1, 1, 3, 3, 7, 7, 12, 14, 20, 22, 32, 34, 45, 51, 63, 69, 87, 93, 112, 124, 144, 156, 184, 196, 225, 245, 275, 295, 335, 355, 396, 426, 468, 498, 552, 582, 637, 679, 735, 777, 847, 889, 960, 1016, 1088, 1144, 1232, 1288, 1377, 1449, 1539, 1611, 1719

Offset: 0

Author

John W. Layman, May 07 2007

Keywords

• nonn

Comments

See A008805 and A049820 for the numbers of partitions p of n such that max(p)-min(p)=1 or 2, respectively.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Crossrefs

Cf. A008805, A049820, A097364, A116685, A117142, A117143.

Programs

Mathematica

np[n_]:=Length[Select[IntegerPartitions[n],Max[#]-Min[#]==3&]]; Array[np,60] (* Harvey P. Dale, Jul 02 2012 *)

Formula

Conjecture. a(1)=0 and, for n>1, a(n+1)=a(n)+d(n), where d(n) is defined as follows: d=0,0,0,1,0 for n=1,...,5 and, for n>5, d(n)=d(n-2)+1 if n=6k or n=6k+4, d(n)=d(n-2) if n=6k+1 or n=6k+3, d(n)=d(n-2)+2Floor[n/6] if n=6k+2 and d(n)=d(n-5) if n=6k+5.

G.f. for number of partitions p of n such that max(p)-min(p) = m is Sum_{k>0} x^(2*k+m)/Product_{i=0..m} (1-x^(k+i)). - Vladeta Jovovic, Jul 04 2007

a(n) = A097364(n,3) = A116685(n,3) = A117143(n) - A117142(n). - Alois P. Heinz, Nov 02 2012

Extensions

More terms from Vladeta Jovovic, Jul 04 2007