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Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 9.
2

%I #7 Apr 03 2014 09:34:04

%S 1,0,1,0,2,0,3,0,5,0,7,0,11,0,15,0,22,0,30,0,41,1,54,2,73,4,94,7,123,

%T 12,157,19,201,30,252,45,318,67,393,97,488,139,598,194,732,269,888,

%U 366,1078,494,1296,658,1558,870,1862,1137,2222,1477,2639,1900,3133

%N Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 9.

%C With offset 90 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -9.

%H Alois P. Heinz, <a href="/A240145/b240145.txt">Table of n, a(n) for n = 81..1000</a>

%F a(n) = [x^n y^9] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

%e a(87) = 3: [23,15,13,11,9,7,5,3,1], [21,17,13,11,9,7,5,3,1], [19,17,15,11,9,7,5,3,1].

%e a(102) = 1: [19,17,15,13,11,9,7,5,3,2,1].

%p b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or

%p abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+

%p `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))

%p end:

%p a:= n-> b(n$2, -9):

%p seq(a(n), n=81..145);

%Y Column k=9 of A240021.

%K nonn

%O 81,5

%A _Alois P. Heinz_, Apr 02 2014