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A301760
Number of rooted twice-partitions of n where the composite rooted partition is constant.
3
1, 1, 2, 4, 7, 11, 17, 24, 34, 46, 63, 82, 109, 140, 183, 233, 298, 376, 479, 598, 753, 938, 1171, 1449, 1797, 2210, 2726, 3342, 4095, 4990, 6088, 7388, 8968, 10843, 13099, 15770, 18975, 22756, 27276, 32603, 38925, 46353, 55158, 65479, 77656, 91904, 108645
OFFSET
1,3
COMMENTS
A rooted partition of n is an integer partition of n - 1. A rooted twice-partition of n is a choice of a rooted partition of each part in a rooted partition of n.
FORMULA
O.g.f.: 1/(1 - x) + Sum_{n > 0} (-1/(1 - x) + Product_{k >= 0} 1/(1 - x^(n * k + 1))).
EXAMPLE
The a(5) = 7 rooted twice-partitions: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().
MATHEMATICA
nn=50;
ser=(1-nn)/(1-x)+Sum[Product[1/(1-x^(d k+1)), {k, 0, nn}], {d, nn}];
CoefficientList[Series[ser, {x, 0, nn}], x]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 26 2018
STATUS
approved