A093637 -id:A093637 - cp's OEIS frontend

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

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

Gus Wiseman, Mar 26 2018

nonn

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

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

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]

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

Gus Wiseman, Mar 26 2018

nonn

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) = 11 rooted twice-partitions:
(7), (1111111),
(6)(), (33)(), (222)(), (111111)(), (11111)(1), (22)(2), (1111)(11),
(1111)(1)(), (111)(11)().

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

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

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

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

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

A308206 G.f.: x * Product_{k>=1} 1/(1 - a(k)*x^k)^k.

Original entry on oeis.org

1, 1, 3, 12, 63, 396, 2926, 24497, 229757, 2377153, 26917186, 330804783, 4387399275, 62455948949, 950123048257, 15384516283921, 264229711285878, 4798448004296966, 91878671010619078, 1850134691327469413, 39088537892778891963, 864610314507158356377

Offset: 1

Ilya Gutkovskiy, May 15 2019

nonn

Cf. A093637, A308204, A308205, A308207.

a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - a[k] x^k)^k, {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 22}]
a[n_] := a[n] = Sum[Sum[d^2 a[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]/(n - 1); a[1] = 1; Table[a[n], {n, 1, 22}]

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d^2*a(d)^(k/d) ) * a(n-k+1).

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

Gus Wiseman, Mar 26 2018

nonn

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

			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)().

Andrew Howroyd, Table of n, a(n) for n = 1..500

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

nn=50;
ser=x*Product[1+PartitionsP[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

seq(n)={Vec(prod(k=1, n-1, 1 + numbpart(k-1)*x^k + O(x^n)))} \\ Andrew Howroyd, Aug 29 2018

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

Gus Wiseman, Mar 26 2018

nonn

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

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

Andrew Howroyd, Table of n, a(n) for n = 1..500

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

nn=50;
ser=x*Product[1/(1-PartitionsQ[n-1]x^n),{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

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

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

Gus Wiseman, Mar 26 2018

nonn

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

			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)().

Andrew Howroyd, Table of n, a(n) for n = 1..500

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

nn=50;
ser=x*Product[1+PartitionsQ[n-1]x^n,{n,nn}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,nn}]

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

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

Gus Wiseman, Mar 26 2018

nonn

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

			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).

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

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

Table[If[n===1,1,Sum[If[d===n-1,1,DivisorSigma[0,(n-1)/d-1]]^d,{d,Divisors[n-1]}]],{n,50}]

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

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}]

A345973 G.f.: x + x^2 / Product_{n>=1} (1 - a(n)*x^n).

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 20, 36, 73, 138, 281, 549, 1136, 2263, 4705, 9553, 20015, 41096, 86643, 179638, 380701, 795892, 1693003, 3562217, 7612680, 16099538, 34505797, 73345831, 157678081, 336419942, 725236780, 1552662599, 3354979195, 7205601904, 15600414855, 33594465666

Offset: 1

Ilya Gutkovskiy, Jun 30 2021

nonn

David Callan, A Combinatorial Interpretation for Sequence A345973 in OEIS, arXiv:2108.04969 [math.CO], 2021.

Cf. A007562, A032307, A093637.

a:= proc(n) option remember; `if`(n<3, 1, add(a(n-k)*add(d*
      a(d)^(k/d), d=numtheory[divisors](k)), k=1..n-2)/(n-2))
    end:
seq(a(n), n=1..37);  # Alois P. Heinz, Jul 01 2021

a[n_] := a[n] = SeriesCoefficient[x + x^2/Product[(1 - a[k] x^k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 37}]
a[1] = a[2] = 1; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d a[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 37}]

G.f.: x + x^2 * exp(Sum_{n>=1} Sum_{k>=1} a(n)^k * x^(n*k) / k).

a(n+2) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d)^(k/d) ) * a(n-k+2).

A093639 Self-convolution forms A093638.

Original entry on oeis.org

1, 1, 3, 10, 40, 163, 738, 3308, 15767, 74784, 368717, 1805773, 9113860, 45683811, 234272577, 1194970083, 6201864043, 32034420492, 167941221248, 876209929880, 4627819824850, 24348860106096, 129422692783990, 685425678409977

Offset: 0

Paul D. Hanna, Apr 07 2004

nonn

Cf. A093637, A093638.

G.f.: A(x) = Product_{n>=0} 1/(1-A093638(n)*x^(n+1)).

cp's OEIS Frontend

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

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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).

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A308206 G.f.: x * Product_{k>=1} 1/(1 - a(k)*x^k)^k.

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

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

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

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

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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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