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.

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A301469 Signed recurrence over enriched r-trees: a(n) = 2 * (-1)^n + Sum_y Product_{i in y} a(y) where the sum is over all integer partitions of n - 1.

Original entry on oeis.org

2, -1, 1, 0, 0, 1, 0, 1, 1, 1, 2, 3, 3, 6, 7, 11, 17, 23, 35, 53, 75, 119, 173, 264, 398, 603, 911, 1411, 2114, 3279, 4977, 7696, 11760, 18253, 27909, 43451, 66675, 103945, 160096, 249904, 385876, 603107, 933474, 1461967, 2266384, 3553167, 5521053, 8664117, 13485744
Offset: 0

Views

Author

Gus Wiseman, Mar 21 2018

Keywords

Crossrefs

Cf. A032305, A055277, A093637, A127524, A196545, A220418, A273866, A273873, A289501, A290261, A290973, A301342-A301345, A301364-A301368, A301422, A301462, A301467, A301470.

Programs

  • Mathematica
    a[n_]:=a[n]=2(-1)^n+Sum[Times@@a/@y,{y,IntegerPartitions[n-1]}];
Array[a,30]

Formula

O.g.f.: 2/(1 + x) + x Product_{i > 0} 1/(1 - a(i) x^i).
a(n) = Sum_t 2^k * (-1)^w where the sum is over all enriched r-trees of size n, k is the number of leaves, and w is the sum of leaves.

A301750 Number of rooted twice-partitions of n where the composite rooted partition is strict.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 18, 29, 42, 61, 86, 127, 181, 257, 352, 489, 668, 935, 1270, 1730, 2312, 3101, 4112, 5533, 7345, 9742, 12785, 16793, 21821, 28452, 36908, 48108, 62198, 80337, 103081, 132372, 168805, 215247, 273678
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(8) = 18 rooted twice-partitions where the composite rooted partition is strict:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (3)(2),
(4)()(), (31)()(), (3)(1)(),
(3)()()(), (21)()()(), (2)(1)()(),
(2)()()()(),
(1)()()()()(),
()()()()()()().

Crossrefs

Cf. A002865, A063834, A093637, A127524, A279790, A294788, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Join@@#&]//Length,{n,30}]

A301760 Number of rooted twice-partitions of n where the composite rooted partition is constant.

Original entry on oeis.org

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

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(5) = 7 rooted twice-partitions: (3), (111), (2)(), (11)(), (1)(1), (1)()(), ()()()().

Crossrefs

Cf. A002865, A047968, A063834, A093637, A295924, A301422, A301462, A301467, A301480, A301706.

Programs

  • 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]

Formula

O.g.f.: 1/(1 - x) + Sum_{n > 0} (-1/(1 - x) + Product_{k >= 0} 1/(1 - x^(n * k + 1))).

A301766 Number of rooted twice-partitions of n where the first rooted partition is strict and the composite rooted partition is constant, i.e., of type (R,Q,R).

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 7, 9, 11, 13, 16, 19, 22, 26, 32, 36, 42, 52, 59, 66, 79, 93, 108, 125, 141, 162, 192, 222, 248, 285, 331, 375, 430, 492, 555, 632, 719, 816, 929, 1051, 1177, 1327, 1510, 1701, 1908, 2146, 2408, 2705, 3035, 3388, 3792, 4257, 4751, 5284, 5894
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) = 11 rooted twice-partitions:
(7), (1111111),
(6)(), (33)(), (222)(), (111111)(), (11111)(1), (22)(2), (1111)(11),
(1111)(1)(), (111)(11)().

Links

Crossrefs

Cf. A002865, A032305, A047966, A063834, A093637, A296134, A300383, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    twirtns[n_]:=Join@@Table[Tuples[IntegerPartitions[#-1]&/@ptn],{ptn,IntegerPartitions[n-1]}];
Table[Select[twirtns[n],UnsameQ@@Total/@#&&SameQ@@Join@@#&]//Length,{n,20}]
  • PARI
    a(n)=if(n<3, 1, sum(k=1, n-2, polcoef(prod(j=0, (n-2)\k, 1 + x^(j*k + 1) + O(x^n)), n-1))) \\ Andrew Howroyd, Aug 26 2018

Extensions

Terms a(26) and beyond from Andrew Howroyd, Aug 26 2018

A301751 Number of ways to choose a rooted partition of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 3, 5, 10, 17, 32, 54, 100, 166, 289, 494, 840, 1393, 2400, 3931, 6498, 10861, 17728, 28863, 47557, 77042, 123881, 201172, 322459, 517032, 827993, 1316064, 2084632, 3328204, 5236828, 8247676, 13005652, 20417628, 31934709, 49970815, 77789059, 121144373
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) = 17 rooted twice-partitions:
(5), (41), (32), (311), (221), (2111), (11111),
(4)(), (31)(), (22)(), (211)(), (1111)(), (3)(1), (21)(1), (111)(1),
(2)(1)(), (11)(1)().

Links

Crossrefs

Cf. A002865, A032305, A063834, A093637, A271619, A281113, A296118, A300383, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    nn=50;
ser=x*Product[1+PartitionsP[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    seq(n)={Vec(prod(k=1, n-1, 1 + numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

Formula

O.g.f.: x * Product_{n > 0} (1 + A000041(n-1) x^n).

A301753 Number of ways to choose a strict rooted partition of each part in a rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 25, 43, 66, 108, 166, 269, 408, 643, 975, 1517, 2277, 3497, 5223, 7936, 11803, 17736, 26219, 39174, 57594, 85299, 124957, 183987, 268158, 392685, 569987, 830282, 1200843, 1740422, 2507823, 3620550, 5197885, 7472229, 10694865, 15319700
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) = 16 rooted twice-partitions:
(5), (32), (41),
(2)(2), (3)(1), (4)(), (21)(1), (31)(),
(1)(1)(1), (2)(1)(), (3)()(), (21)()(),
(1)(1)()(), (2)()()(),
(1)()()()(),
()()()()()().

Links

Crossrefs

Cf. A002865, A032305, A063834, A093637, A270995, A281113, A296119, A301422, A301462, A301467, A301480, A301706, A301751.

Programs

  • Mathematica
    nn=50;
ser=x*Product[1/(1-PartitionsQ[n-1]x^n),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    seq(n)={my(u=Vec(prod(k=1, n-1, 1 + x^k + O(x^n)))); Vec(1/prod(k=1, n-1, 1 - u[k]*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

Formula

O.g.f.: x * Product_{n > 0} 1/(1 - A000009(n-1) x^n).

A301754 Number of ways to choose a strict rooted partition of each part in a strict rooted partition of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 18, 29, 44, 67, 100, 150, 217, 326, 470, 690, 1011, 1463, 2099, 3049, 4355, 6214, 8886, 12632, 17885, 25377, 35763, 50252, 70942, 99246, 138600, 193912, 270286, 375471, 522224, 723010, 1000435, 1383002, 1907724, 2624492, 3613885
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) = 13 rooted twice-partitions:
(6), (51), (42), (321),
(5)(), (41)(), (32)(), (4)(1), (31)(1), (3)(2), (21)(2),
(3)(1)(), (21)(1)().

Links

Crossrefs

Cf. A002865, A032305, A063834, A093637, A196545, A279785, A296120, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    nn=50;
ser=x*Product[1+PartitionsQ[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]
  • PARI
    seq(n)={my(u=Vec(prod(k=1, n-1, 1 + x^k + O(x^n)))); Vec(prod(k=1, n-1, 1 + u[k]*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

Formula

O.g.f.: x * Product_{n > 0} (1 + A000009(n-1) x^n).

A301763 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 5, 8, 13, 14, 5, 32, 7, 20, 64, 26, 6, 92, 7, 126, 199, 22, 5, 352, 252, 41, 581, 394, 7, 1832, 9, 292, 2119, 31, 3216, 4946, 10, 40, 8413, 7708, 9, 20656, 9, 2324, 53546, 24, 5, 70040, 16395, 59361, 131204, 9503, 7, 266780, 178180, 82086
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) = 8 rooted twice-partitions: (5), (11111), (2)(2), (2)(11), (11)(2), (11)(11), (1)(1)(1), ()()()()()().
The a(15) = 20 rooted twice-partitions:
()()()()()()()()()()()()()(),
(1)(1)(1)(1)(1)(1)(1), (111111)(111111), (1111111111111),
(111111)(222), (222)(111111), (222)(222),
(111111)(33), (222)(33), (33)(111111), (33)(222), (33)(33),
(111111)(6), (222)(6), (33)(6), (6)(111111), (6)(222), (6)(33), (6)(6),
(13).

Links

Crossrefs

Cf. A000005, A002865, A047968, A063834, A093637, A127524, A295924, A300383, A301422, A301462, A301467, A301480, A301706.

Programs

  • Mathematica
    Table[If[n===1,1,Sum[If[d===n-1,1,DivisorSigma[0,(n-1)/d-1]]^d,{d,Divisors[n-1]}]],{n,50}]
  • PARI
    a(n)=if(n==1, 1, sumdiv(n-1, d, if(d==n-1, 1, numdiv((n-1)/d-1)^d))) \\ Andrew Howroyd, Aug 26 2018

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}]
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