A104460 -id:A104460 - cp's OEIS frontend

A109092 Number of hierarchical orderings for n labeled elements with 2 possible classes A and B for levels l>=2. Labeled analog of A104460.

1, 6, 53, 619, 8972, 155067, 3109269, 70893872, 1810283331, 51151579619, 1583934062306, 53322541667501, 1938521128765093, 75673000809822670, 3156390306304019025, 140076451219218605087, 6589244960448222899044, 327461842184597424792623, 17141751726301435708168665

Offset: 1

Thomas Wieder, Jun 18 2005

nonn

			Let | denote a separator among different hierarchies of the hierarchical ordering. Let : denote a separator between levels in a hierarchy.
Furthermore, let a[1], a[2],... denote labeled elements.
An element a[i] will be written as a[i,A] if it falls into class A and as a[i,B] if it falls into class B. Note that at level l=1 no classes appear.
Then a(2) = 6 because a[1]a[2], a[1]|a[2], a[1]:a[2,A], a[2]:a[1,A], a[1]:a[2,B], a[2]:a[1,B].

Alois P. Heinz, Table of n, a(n) for n = 1..140
Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 2.
Norihiro Nakashima, Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Cf. A075729, A104460.

with(combstruct): A109092 := [T, {T=Set(Sequence(S,card>=1)), S=Sequence(U,card>=1), U=Set(Z,card>=1)},labeled]; seq(count(A109092, size=j), j=1..20);

With[{nn=20},CoefficientList[Series[Exp[-(Exp[x]-1)/(-3+2Exp[x])],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 16 2016 *)

G.f.: exp(-(exp(z)-1)/(-3+2*exp(z))).

A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

Original entry on oeis.org

1, 1, 5, 21, 95, 415, 1851, 8155, 36030, 158510, 696502, 3052966, 13359230, 58346206, 254405630, 1107479694, 4813850699, 20894227355, 90567536543, 392066476815, 1695180397145, 7320927664713, 31581573600685, 136094434672509, 585876330191950, 2519701493092958

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144068, A343350, A343351, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*4^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 12 2021

nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^(4^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

a(n) ~ exp(sqrt(n) - 1/8 + c/4) * 2^(2*n - 3/2) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (4^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343350 Expansion of Product_{k>=1} 1 / (1 - x^k)^(5^(k-1)).

Original entry on oeis.org

1, 1, 6, 31, 171, 921, 5031, 27281, 148101, 801901, 4336902, 23415777, 126254962, 679805112, 3655679442, 19634501447, 105334380517, 564471596667, 3021754455157, 16160029793032, 86339725851558, 460874548444683, 2457961986888773, 13097958657023523, 69740119667456018

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144069, A343349, A343351, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*5^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 12 2021

nmax = 24; CoefficientList[Series[Product[1/(1 - x^k)^(5^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 5^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]

a(n) ~ exp(2*sqrt(n/5) - 1/10 + c/5) * 5^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (5^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343351 Expansion of Product_{k>=1} 1 / (1 - x^k)^(6^(k-1)).

Original entry on oeis.org

1, 1, 7, 43, 280, 1792, 11586, 74550, 479892, 3083640, 19794678, 126908502, 812761299, 5199586119, 33230586285, 212172173565, 1353444677529, 8626044781761, 54931168743703, 349524243121795, 2222294161109422, 14119034725444774, 89639674321304392, 568720801952770012

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144070, A343349, A343350, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*6^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(6^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 6^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

a(n) ~ exp(sqrt(2*n/3) - 1/12 + c/6) * 6^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (6^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343352 Expansion of Product_{k>=1} 1 / (1 - x^k)^(7^(k-1)).

Original entry on oeis.org

1, 1, 8, 57, 428, 3172, 23689, 176324, 1312550, 9757798, 72480269, 537854094, 3987751860, 29540543908, 218652961074, 1617159619805, 11951595353413, 88264810625245, 651404299886762, 4804261815210433, 35410065096578748, 260832137791524693, 1920169120639498017, 14127684273966098698

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144071, A343349, A343350, A343351, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*7^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 12 2021

nmax = 23; CoefficientList[Series[Product[1/(1 - x^k)^(7^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 7^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 23}]

a(n) ~ exp(2*sqrt(n/7) - 1/14 + c/7) * 7^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (7^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343353 Expansion of Product_{k>=1} 1 / (1 - x^k)^(8^(k-1)).

Original entry on oeis.org

1, 1, 9, 73, 621, 5229, 44293, 374277, 3162447, 26694159, 225163687, 1897751079, 15983278059, 134519816427, 1131395821587, 9509592524371, 79880259426102, 670590654977718, 5626336598011078, 47179486350900358, 395410837699366686, 3312225325409475038, 27731588831310844302

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144072, A343349, A343350, A343351, A343352, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*8^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 12 2021

nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^(8^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 8^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 22}]

a(n) ~ exp(sqrt(n/2) - 1/16 + c/8) * 2^(3*n - 7/4) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (8^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343354 Expansion of Product_{k>=1} 1 / (1 - x^k)^(9^(k-1)).

Original entry on oeis.org

1, 1, 10, 91, 865, 8155, 77251, 730435, 6905560, 65233120, 615847378, 5810270782, 54784324495, 516250199827, 4862041512625, 45765734635702, 430560567351208, 4048630897384450, 38051334554031551, 357459295903931045, 3356488167698692226, 31503001136703776561

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144073, A343349, A343350, A343351, A343352, A343353, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*9^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(9^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 9^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

a(n) ~ exp(2*sqrt(n/9) - 1/18 + c/9) * 9^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (9^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343355 Expansion of Product_{k>=1} 1 / (1 - x^k)^(10^(k-1)).

Original entry on oeis.org

1, 1, 11, 111, 1166, 12166, 127436, 1332936, 13939651, 145683351, 1521743103, 15886781603, 165770328383, 1728861822083, 18022063489023, 187778810866043, 1955660195168328, 20358764860253028, 211849198103034998, 2203562708619192998, 22911457758236641451, 238129937419462634151

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

In general, if m > 1 and g.f. = Product_{k>=1} 1 / (1 - x^k)^(m^(k-1)), then a(n, m) ~ exp(2*sqrt(n/m) - 1/(2*m) + c(m)/m) * m^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c(m) = Sum_{j>=2} 1/(j * (m^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

Cf. A034691, A104460, A292837, A343349, A343350, A343351, A343352, A343353, A343354.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*10^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..21);  # Alois P. Heinz, Apr 12 2021

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(10^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 10^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 21}]

a(n) ~ exp(sqrt(2*n/5) - 1/20 + c/10) * 10^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (10^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343360 Expansion of Product_{k>=1} (1 + x^k)^(3^(k-1)).

Original entry on oeis.org

1, 1, 3, 12, 39, 138, 469, 1603, 5427, 18372, 61869, 207909, 696537, 2328039, 7762266, 25826142, 85749969, 284171598, 940027872, 3104280885, 10234808334, 33692547249, 110753171784, 363561071175, 1191860487561, 3902350627434, 12761565487173, 41685086306917, 136012008938158

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A098407, A104460, A256142, A343361, A343362, A343363, A343364, A343365, A343366.

h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1)*binomial(3^(i-1), j), j=0..n/i)))
    end:
a:= n-> h(n$2):
seq(a(n), n=0..28);  # Alois P. Heinz, Apr 12 2021

nmax = 28; CoefficientList[Series[Product[(1 + x^k)^(3^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[(-1)^(k/d + 1) d 3^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 28}]

seq(n)={Vec(prod(k=1, n, (1 + x^k + O(x*x^n))^(3^(k-1))))} \\ Andrew Howroyd, Apr 12 2021

a(n) ~ exp(2*sqrt(n/3) - 1/6 - c/3) * 3^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} (-1)^j / (j * (3^(j-1) - 1)). - Vaclav Kotesovec, Apr 13 2021

A104500 Number of different groupings among the hierarchical orderings of n unlabeled elements.

Original entry on oeis.org

1, 4, 11, 35, 98, 294, 832, 2401, 6774, 19137, 53466, 148994, 412233, 1136383, 3116654, 8515706, 23172455, 62836916, 169801824, 457406173, 1228382159, 3289493887, 8784935160, 23400668297, 62179339101, 164832960183, 435978612329, 1150673925933, 3030701471118

Offset: 1

Thomas Wieder, Mar 11 2005

nonn

			Let * denote an element, let : denote separator among different levels within a hierarchy, let | denote a separator between different hierarchies. Furthermore, the braces {} indicate a group. For n=3 one has a(3) = 11 because
{***}, {*|*|*}, {*}{*}{*}, {*:*:*}, {*:**}, {*|**}, {*:*|*}, {*:*}{*}, {*|*}{*}, {**:*}, {*}{**}.

Alois P. Heinz, Table of n, a(n) for n = 1..800
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Cf. A034691, A104460, A034899.

etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=numtheory[divisors](j)) *b(n-j), j=1..n)/n) end end: b:= etr(n-> 2^(n-1)): a:= etr(b): seq(a(n), n=1..30); # Alois P. Heinz, Apr 21 2012

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b = etr[Function[{n}, 2^(n-1)]]; a = etr[b]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

Euler transform of 1, 3, 7, 18, 42, 104, 244, 585, 1373, ... = A034691.

More terms from Alois P. Heinz, Apr 21 2012

cp's OEIS Frontend

A109092 Number of hierarchical orderings for n labeled elements with 2 possible classes A and B for levels l>=2. Labeled analog of A104460.

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A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

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A343350 Expansion of Product_{k>=1} 1 / (1 - x^k)^(5^(k-1)).

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A343351 Expansion of Product_{k>=1} 1 / (1 - x^k)^(6^(k-1)).

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A343352 Expansion of Product_{k>=1} 1 / (1 - x^k)^(7^(k-1)).

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A343353 Expansion of Product_{k>=1} 1 / (1 - x^k)^(8^(k-1)).

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A343354 Expansion of Product_{k>=1} 1 / (1 - x^k)^(9^(k-1)).

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A343355 Expansion of Product_{k>=1} 1 / (1 - x^k)^(10^(k-1)).

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A343360 Expansion of Product_{k>=1} (1 + x^k)^(3^(k-1)).

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