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A218567
Number of partitions p of n such that max(p)-min(p) = 4.
4
1, 1, 3, 3, 7, 8, 13, 16, 24, 27, 40, 46, 60, 71, 92, 103, 131, 149, 181, 206, 247, 275, 329, 366, 424, 474, 548, 601, 690, 759, 858, 942, 1059, 1152, 1293, 1404, 1555, 1690, 1869, 2013, 2218, 2390, 2614, 2812, 3066, 3282, 3574, 3820, 4131, 4415, 4769, 5071
OFFSET
6,3
LINKS
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.
FORMULA
G.f.: Sum_{k>0} x^(2*k+4)/Product_{j=0..4} (1-x^(k+j)).
a(n) = A097364(n,4) = A116685(n,4) = A194621(n,4) - A194621(n,3) = A218506(n) - A117143(n).
MATHEMATICA
terms = 52; offset = 6; 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[4], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n], _?(#[[1]]-#[[-1]]==4&)], {n, 6, 60}] (* Harvey P. Dale, Jul 10 2021 *)
CROSSREFS
Sequence in context: A178238 A255333 A200792 * A161416 A241637 A241641
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 02 2012
STATUS
approved