A218570 Number of partitions p of n such that max(p)-min(p) = 7.

1, 1, 3, 3, 7, 8, 14, 18, 28, 34, 51, 63, 86, 108, 144, 175, 229, 278, 351, 425, 529, 630, 775, 919, 1109, 1309, 1565, 1827, 2167, 2518, 2952, 3414, 3975, 4563, 5281, 6036, 6931, 7889, 9012, 10200, 11598, 13078, 14785, 16613, 18704, 20925, 23470, 26174, 29229

Offset: 9

Author

Alois P. Heinz, Nov 02 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 9..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 = 49; offset = 9; 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[7], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)

Formula

G.f.: Sum_{k>0} x^(2*k+7)/Product_{j=0..7} (1-x^(k+j)).

a(n) = A097364(n,7) = A116685(n,7) = A194621(n,7) - A194621(n,6) = A218509(n) - A218508(n).