A118808 Number of partitions of n having exactly one part with multiplicity 3.
0, 0, 0, 1, 0, 1, 2, 3, 3, 5, 8, 13, 13, 23, 28, 40, 49, 71, 89, 123, 147, 198, 249, 329, 400, 518, 642, 825, 996, 1265, 1545, 1941, 2340, 2920, 3533, 4357, 5233, 6417, 7717, 9399, 11211, 13591, 16215, 19540, 23189, 27826, 32990, 39392, 46504, 55313, 65200
Offset: 0
Keywords
Examples
a(9)=5 because we have [6,1,1,1],[4,2,1,1,1],[3,3,3],[3,3,1,1,1] and [3,2,2,2].
Programs
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Maple
g:=product((1-x^(3*j)+x^(4*j))/(1-x^j),j=1..70)*sum(x^(3*j)*(1-x^j)/(1-x^(3*j)+x^(4*j)),j=1..70): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..60);
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Mathematica
Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#], 3]==1&]],{n,0,60}] (* Harvey P. Dale, Mar 24 2011 *)
Formula
G.f.=product([1-x^(3j)+x^(4j)]/(1-x^j), j=1..infinity)*sum(x^(3j)*(1-x^j)/[1-x^(3j)+x^(4j)], j=1..infinity).
Comments