A127524 -id:A127524 - cp's OEIS frontend

A301765 Number of rooted twice-partitions of n where the first rooted partition is constant and the composite rooted partition is strict, i.e., of type (Q,R,Q).

1, 1, 2, 2, 3, 3, 4, 5, 8, 7, 11, 11, 19, 16, 27, 23, 42, 33, 63, 47, 87, 71, 119, 90, 195

Offset: 1

Gus Wiseman, Mar 26 2018

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.

			The a(9) = 8 rooted twice-partitions:
(7), (61), (52), (43), (421),
(3)(21), (21)(3),
()()()()()()()().

Cf. A002865, A032305, A063834, A093637, A127524, A279791, A296133, A301422, A301462, A301467, A301480, A301706.

twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],SameQ@@Total/@#&&UnsameQ@@Join@@#&]//Length,{n,20}]

A301761 Number of ways to choose a rooted partition of each part in a constant rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 13, 12, 26, 31, 57, 43, 150, 78, 224, 293, 484, 232, 1190, 386, 2260, 2087, 2558, 1003, 11154, 4701, 7889, 13597, 30041, 3719, 83248, 5605, 95006, 84486, 63506, 251487, 654394, 17978, 169864, 490741, 2290336, 37339, 4079503, 53175, 3979370

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(7) = 13 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(2)(2), (2)(11), (11)(2), (11)(11),
(1)(1)(1),
()()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000041, A002865, A018818, A063834, A093637, A127524, A260685, A271619, A279787, A301422, A301462, A301467, A301480, A301706.

Table[Sum[PartitionsP[n/d-1]^d,{d,Divisors[n]}],{n,50}]

a(n)=if(n==1, 1, sumdiv(n-1, d, numbpart((n-1)/d-1)^d)) \\ Andrew Howroyd, Aug 26 2018

a(n) = Sum_{d | n-1} A000041((n-1)/d-1)^d for n > 1. - Andrew Howroyd, Aug 26 2018

A301768 Number of ways to choose a strict rooted partition of each part in a constant rooted partition of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 5, 11, 8, 14, 11, 32, 16, 36, 32, 70, 33, 104, 47, 168, 130, 178, 90, 521, 155, 369, 383, 902, 223, 1562, 297, 1952, 1392, 1474, 1665, 6297, 669, 2878, 4241, 12401, 1114, 17474, 1427, 19436, 20754, 9971, 2305, 80110, 19295, 51942, 36428

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(9) = 11 rooted twice-partitions:
(7), (61), (52), (43), (421),
(3)(3), (3)(21), (21)(3), (21)(21),
(1)(1)(1)(1),
()()()()()()()().

Cf. A002865, A063834, A093637, A127524, A279788, A296132, A301422, A301462, A301467, A301480, A301706.

Table[Sum[PartitionsQ[n/d-1]^d,{d,Divisors[n]}],{n,50}]

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A301765 Number of rooted twice-partitions of n where the first rooted partition is constant and the composite rooted partition is strict, i.e., of type (Q,R,Q).

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A301761 Number of ways to choose a rooted partition of each part in a constant rooted partition of n.

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A301768 Number of ways to choose a strict rooted partition of each part in a constant rooted partition of n.

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