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

Author

David S. Newman, Dec 27 2010

Keywords

• nonn

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.

Crossrefs

Cf. A098859, A144328.

Programs

Mathematica

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Extensions

More terms from Alois P. Heinz, Jun 25 2011