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

%I A218573 #17 Aug 15 2024 11:39:50
%S A218573 1,1,3,3,7,8,14,18,28,35,53,67,93,119,161,201,267,332,428,531,674,824,
%T A218573 1034,1258,1552,1877,2294,2749,3332,3970,4762,5645,6723,7916,9367,
%U A218573 10974,12894,15036,17571,20381,23696,27370,31652,36416,41926,48029,55071,62860
%N A218573 Number of partitions p of n such that max(p) - min(p) = 10.
%H A218573 Alois P. Heinz, <a href="/A218573/b218573.txt">Table of n, a(n) for n = 12..1000</a>
%H A218573 G. E. Andrews, M. Beck and N. Robbins, <a href="https://arxiv.org/abs/1406.3374">Partitions with fixed differences between largest and smallest parts</a>, arXiv:1406.3374 [math.NT], 2014.
%F A218573 G.f.: Sum_{k>0} x^(2*k+10)/Product_{j=0..10} (1-x^(k+j)).
%F A218573 a(n) = A097364(n,10) = A116685(n,10) = A194621(n,10) - A194621(n,9) = A218512(n) - A218511(n).
%t A218573 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_ *)
%t A218573 Table[Count[IntegerPartitions[n],_?(#[[1]]-#[[-1]]==10&)],{n,12,60}] (* _Harvey P. Dale_, Aug 14 2024 *)
%K A218573 nonn
%O A218573 12,3
%A A218573 _Alois P. Heinz_, Nov 02 2012