A178702 Coefficients of the sum 1+ x/(1-x) + x^2/(1-x^2) + x^3/ ( (1-x)(1-x^2)) + x^4/ ( (1-x)(1-x^3) ) + x^5/ ( (1-x)(1-x^4) ) + x^5 /((1-x^2)(1-x^3)) + x^6/ ( (1-x)(1-x^2)(1-x^3)) + ...

1, 1, 2, 2, 4, 6, 7, 10, 14, 20, 24, 32, 40, 54, 69, 86, 106, 135, 165, 206, 256, 311, 378, 460, 555, 670, 808, 970, 1156, 1380, 1638, 1938, 2296, 2706, 3188, 3752, 4390, 5136, 6000, 6990, 8128, 9444, 10944, 12672, 14659, 16904, 19476, 22420, 25753, 29550, 33873, 38759, 44306

Offset

0,3

Comments

For a given positive integer, n, let S_n be the set of partitions of n into distinct parts where the number of parts is maximal for that n. For example, for n=6, the set S_6 consists of just one such partition S_6={1,2,3}. Similarly, for n=7, S_7={1,2,4}, But for n=8, S_8 will contain two partitions S_8= { {1,2,5}, {1,3,4} }. Now form the sum whose general term is x^n divided by the product (1-x^(p_1))...(1-x^(p_i)) where the p's come from the partitions in S_n. The sequence is the sequence of coefficients of this sum.

This sequence is an upper bound for A098859.

The initial values of the sequence were verified by several members of the Sequence Fans mailing list.

|S_n| = A144328(n) (if the offset of the latter is changed to 0). - Benoit Jubin, Dec 13 2010.

Links

Table of n, a(n) for n=0..52.

Mathematica

<<Combinatorica`
f[set_]:=x^Total[set]/Product[1-x^set[[i]], {i, 1, Length[set]}]
gf=1;
For[n=1, n<=20, n++, Print["n= ", n];
For[i=1, i<=Length[Partitions[n]], i++,
t=Tally[Partitions[n][[i]]];
sum=0;
For[j=1, j<=Length[t], j++,
If[t[[j]][[2]]==1, sum++]];
If[sum==Length[t]
&&Length[t]==Floor[(-1+Sqrt[1+8 n])/2],
gf=gf+f[Partitions[n][[i]]]; Print[CoefficientList[Series[gf, {x, 0, n}], x]];
]]]

Crossrefs

Cf. A098859, A144328.

Sequence in context: A145804 A227560 A046639 * A377467 A265992 A089284

Adjacent sequences: A178699 A178700 A178701 * A178703 A178704 A178705

Keyword

nonn

Author

David S. Newman, Dec 27 2010

Extensions

More terms from Alois P. Heinz, Jun 25 2011

Status

approved