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a(n) ~ exp(4*Pi*sqrt(2*n/57)) * 2^(3/4) * cos(9*Pi/38) / (3^(1/4) * 19^(3/4) * n^(3/4)). - Vaclav Kotesovec, May 10 2018

nmax = 60; Rest[CoefficientList[Series[Product[(1 - x^(19*k))*(1 - x^(19*k+ 5-19))*(1 - x^(19*k- 5))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 10 2018 *)

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editing

Olivier Gerard (olivier.gerard(AT)gmail.com)

Olivier Gérard

nonn,easy,new

Olivier Gerard (ogerardolivier.gerard(AT)ext.jussieugmail.frcom)

Number of partitions of n into parts not of the form 19k, 19k+5 or 19k-5. Also number of partitions with at most 4 parts of size 1 and differences between parts at distance 8 are greater than 1.

nonn,easy,new

Partitions in Number of partitions of n into parts not of the kind form 19k, 19k+5 or 19k-5. Also partitions with at most 4 parts of size 1 and differences between parts at distance 8 are greater than 1.

nonn,easy,new

nonn,easy,part,new

Olivier Gerard (ogerard@(AT)ext.jussieu.fr)

Partitions in parts not of the kind 19k, 19k+5 or 19k-5. Also partitions with at most 4 parts of size 1 and differences between parts at distance 8 are greater than 1.

1, 2, 3, 5, 6, 10, 13, 19, 25, 35, 45, 62, 79, 104, 133, 173, 217, 279, 348, 440, 546, 683, 840, 1043, 1275, 1567, 1907, 2328, 2815, 3416, 4111, 4957, 5940, 7125, 8498, 10148, 12055, 14327, 16959, 20075, 23673, 27920, 32816, 38562, 45185, 52923

1,2

Case k=9,i=5 of Gordon Theorem.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

nonn,easy,part

Olivier Gerard (ogerard@ext.jussieu.fr)

approved