A218573 Number of partitions p of n such that max(p) - min(p) = 10.

1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 53, 67, 93, 119, 161, 201, 267, 332, 428, 531, 674, 824, 1034, 1258, 1552, 1877, 2294, 2749, 3332, 3970, 4762, 5645, 6723, 7916, 9367, 10974, 12894, 15036, 17571, 20381, 23696, 27370, 31652, 36416, 41926, 48029, 55071, 62860

Offset: 12

Author

Alois P. Heinz, Nov 02 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 12..1000
• G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

Programs

Mathematica

terms = 48; offset = 12; max = terms + offset; s[k0_ /; k0 > 0] := Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; Drop[s[10], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(#[[1]]-#[[-1]]==10&)],{n,12,60}] (* _Harvey P. Dale, Aug 14 2024 *)

Formula

G.f.: Sum_{k>0} x^(2*k+10)/Product_{j=0..10} (1-x^(k+j)).

a(n) = A097364(n,10) = A116685(n,10) = A194621(n,10) - A194621(n,9) = A218512(n) - A218511(n).