cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

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).

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Mar 26 2018

Keywords

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.

Examples

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

Crossrefs

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

Programs

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

A301756 Number of ways to choose disjoint strict rooted partitions of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 10, 15, 22, 30, 42, 60, 85, 114, 155, 206, 286, 394, 524, 683, 910, 1187, 1564, 2090, 2751, 3543, 4606, 5917, 7598, 9771, 12651, 16260, 20822, 26421, 33525, 42463, 53594, 67337, 85299
Offset: 1

Views

Author

Gus Wiseman, Mar 26 2018

Keywords

Comments

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

Examples

			The a(8) = 10 rooted twice-partitions:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (3)(2),
(3)(1)().

Crossrefs

Cf. A002865, A032305, A063834, A093637, A275780, A279375, A294786, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],And[UnsameQ@@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

Views

Author

Gus Wiseman, Mar 26 2018

Keywords

Comments

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

Examples

			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),
()()()()()().

Links

Crossrefs

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

Programs

  • Mathematica
    Table[Sum[PartitionsP[n/d-1]^d,{d,Divisors[n]}],{n,50}]
  • PARI
    a(n)=if(n==1, 1, sumdiv(n-1, d, numbpart((n-1)/d-1)^d)) \\ Andrew Howroyd, Aug 26 2018

Formula

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

A301762 Number of ways to choose a constant rooted partition of each part in a rooted partition of n.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 21, 34, 55, 90, 143, 220, 347, 528, 805, 1226, 1831, 2719, 4048, 5940, 8710, 12714, 18403, 26529, 38220, 54679, 77899, 110810, 156848, 221181, 311635, 436705, 610597, 852125, 1184928, 1644136, 2276551, 3142523, 4328960, 5953523, 8167209
Offset: 1

Views

Author

Gus Wiseman, Mar 26 2018

Keywords

Comments

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

Examples

			The a(5) = 7 rooted twice-partitions where the latter rooted partitions are constant: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().

Crossrefs

Cf. A002865, A063834, A093637, A279784, A295935, A300383, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    Table[Sum[Product[If[k===1,1,DivisorSigma[0,k-1]],{k,ptn}],{ptn,IntegerPartitions[n-1]}],{n,20}]

Formula

O.g.f.: Product_{n>0} 1/(1 - d(n-1) x^n) where d(n) = A000005(n) and d(0) = 1.

A301767 Number of ways to choose a constant rooted partition of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 9, 15, 21, 32, 45, 59, 89, 117, 162, 225, 309, 394, 538, 707, 929, 1240, 1613, 2055, 2677, 3517, 4439, 5724, 7288, 9222, 11671, 14809, 18480, 23226, 29138, 36501, 45373, 56438, 69920, 86426, 106715, 131171, 161428, 197717, 242301, 295888
Offset: 1

Views

Author

Gus Wiseman, Mar 26 2018

Keywords

Comments

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

Examples

			The a(7) = 9 rooted twice-partitions:
(5), (11111),
(4)(), (22)(), (1111)(), (3)(1), (111)(1),
(2)(1)(), (11)(1)().

Crossrefs

Cf. A002865, A032305, A063834, A093637, A279786, A296131, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    Table[Sum[Product[If[k===1,1,DivisorSigma[0,k-1]],{k,ptn}],{ptn,Select[IntegerPartitions[n-1],UnsameQ@@#&]}],{n,50}]

Formula

O.g.f.: Product_{n>0} (1 + d(n-1) x^n) where d(n) = A000005(n) and d(0) = 1.

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

Views

Author

Gus Wiseman, Mar 26 2018

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

  • Mathematica
    Table[Sum[PartitionsQ[n/d-1]^d,{d,Divisors[n]}],{n,50}]
Previous Showing 11-16 of 16 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.