A075729 -id:A075729 - cp's OEIS frontend

A354291 Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(4 - 3*exp(x)) = e.g.f. for A032033.

1, 3, 30, 435, 8211, 190056, 5196099, 163541055, 5815620696, 230350071189, 10048990989747, 478467217544322, 24678559536271581, 1370217125170670367, 81457311857722336614, 5160975525978898855143, 347090708803947931122807, 24690132231344937537382560

Offset: 0

Seiichi Manyama, May 23 2022

nonn

Cf. A075729, A354290.

Cf. A032033, A354287, A354289.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*(exp(x)-1)/(4-3*exp(x)))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 3^k*k!*stirling(j, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A032033(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 3^k * A000262(k) * Stirling2(n,k).
a(n) ~ exp(-7/8 - n + 1/(8*log(4/3)) + sqrt(n/log(4/3))) * n^(n - 1/4) / (2*log(4/3)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022

A088729 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

Original entry on oeis.org

1, 3, 1, 13, 9, 1, 75, 79, 18, 1, 541, 765, 265, 30, 1, 4683, 8311, 3870, 665, 45, 1, 47293, 100989, 59101, 13650, 1400, 63, 1, 545835, 1362439, 960498, 278901, 38430, 2618, 84, 1, 7087261, 20246445, 16700545, 5844510, 1012431, 92610, 4494, 108, 1

Offset: 1

Vladeta Jovovic, Nov 22 2003

Also the Bell transform of A000670(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Also the number of k-dimensional flats of the n-dimensional Catalan arrangement. - Shuhei Tsujie, May 05 2019

N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A000670(first column), A075729(row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> add(combinat:-eulerian1(n+1, k)*2^k, k=0..n+1), 9); # Peter Luschny, Jan 26 2016

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, HurwitzLerchPhi[1/2, -n-1, 0]/2], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 27 2018, after Peter Luschny *)

E.g.f.: exp((exp(x)-1)*y/(2-exp(x))).

A099880 Number of preferential arrangements (or simple hierarchies) of 2*n labeled elements with two kinds of elements (where each kind has n elements).

Original entry on oeis.org

1, 2, 18, 260, 5250, 136332, 4327092, 162309576, 7024896450, 344582629820, 18890850749628, 1144656941236536, 75963981061424820, 5479642938171428600, 426894499408073653800, 35720957482170932284560, 3195135789350678836128450, 304234032845362459798904220

Offset: 0

Thomas Wieder, Nov 02 2004

nonn

The unlabeled case seems to be given by A003480, which can be generated by the following combstruct command: SeqUnionU := [S, {S=Sequence(Set(U,card>=1), card>=1), U=Union(a,b), a=Atom, b=Atom},unlabeled]; [seq(count(SeqUnionU, size=n), n=0..20)]; .

			Let a[1], a[2],...,a[n] and b[1],b[2],...,b[n] denote two kinds "a" and "b" of labeled elements where each kind as n elements in total.
Let ":" denote a level, e.g., if the elements a[1] and a[2] are on level L=1 and the element b[1] is on level L=2 then a[1]a[2]:b[1] is a preferrential arrangement (a simple hierarchy) with two levels.
Then for n=2 we have a(2) = 18 arrangements: a[1]a[2]; a[1]:a[2]; a[2]:a[1]; a[1]b[1]; a[1]:b[1]; b[1]:a[1]; a[1]b[2]; a[1]:b[2]; b[2]:a[1]; a[2]b[1]; a[2]:b[1]; b[1]:a[2]; a[2]b[2]; a[2]:b[2]; b[2]:a[2]; b[1]b[2]; b[1]:b[2]; b[2]:b[1].

Alois P. Heinz, Table of n, a(n) for n = 0..348

Cf. A000670, A075729, A003480, A154921.

a:=n-> add(binomial(2*n, n)*(Stirling2(n, k))*k!, k=0..n): seq(a(n), n=0..16); # Zerinvary Lajos, Oct 19 2006
# second Maple program:
b:= proc(n) b(n):= `if`(n=0, 1, add(b(n-j)/j!, j=1..n)) end:
a:= n-> b(n)*(2*n)!/n!:
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 03 2019

f[n_] := Sum[l! StirlingS2[n, l] Binomial[2n, n], {l, n}]; Table[ f[n], {n, 0, 16}] (* Robert G. Wilson v, Nov 04 2004 *)

a(n) = binomial(2*n, n) * Sum_{k=0..n} k! * Stirling2(n, k).
a(n) = binomial(2*n, n) * A000670(n).
a(n) = A154921(2n,n). - Mats Granvik, Feb 07 2009

More terms from Robert G. Wilson v, Nov 04 2004

a(0) corrected and edited by Alois P. Heinz, Feb 03 2019

A109509 Number of hierarchical orderings with at least 2 elements on each level for n unlabeled elements. Unlabeled analog of A097236.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 14, 28, 47, 88, 152, 279, 486, 876, 1539, 2744, 4824, 8551, 15023, 26503, 46509, 81747, 143210, 251007, 438915, 767403, 1339487, 2336955, 4071906, 7090589, 12333894, 21440241, 37235775, 64624267, 112067176, 194209732, 336313393, 582019000

Offset: 0

Thomas Wieder, Jun 30 2005

nonn

A109509 is the Euler transform of the right-shifted Fibonacci numbers A000045.

			Let * denote an unlabeled element.
Let | denote a delimiter between two hierarchies. E.g., for n=3 we have in **|* two hierarchies (each with one level only).
Let : denote a higher level (within a single hierarchy). E.g., for n=6 we have in ***:**:* a single hierarchy distributed over three levels.
Then a(5) = 4 because we have *****, ***:**, **:***, **|***.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Cf. A000045, A075729, A097236, A166861.

SeqSetSetxU := [T, {T=Set(S),S=Sequence(U,card>=1),U=Set(Z,card>=2)},unlabeled]; seq(count(SeqSetSetxU,size=j),j=1..25); # where x is an integer 1, 2, 3,... # x=2 gives 2 individuals per level.

CoefficientList[Series[Product[1/(1-x^k)^Fibonacci[k-1], {k, 1, 40}], {x, 0, 40}], x] (* Vaclav Kotesovec, Aug 06 2015 *)

ET(v)=Vec(prod(k=1,#v,1/(1-x^k+x*O(x^#v))^v[k]))
ET(vector(40,n,fibonacci(n-1)))

a(n) ~ phi^(n-1/4) / (2 * sqrt(Pi) * 5^(1/8) * n^(3/4)) * exp(phi/10 - 1/2 + 2*5^(-1/4)*sqrt(n/phi) + s), where s = Sum_{k>=2} 1/((phi^(2*k) - phi^k - 1)*k) = 0.189744799982532613329750744326543900883761701983311537716143669... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 06 2015

Edited with more terms from Franklin T. Adams-Watters, Oct 21 2009

A110045 Number of hierarchical orderings ("societies") of n unlabeled elements ("individuals") with at least two occupied levels.

Original entry on oeis.org

1, 0, 1, 3, 8, 18, 45, 102, 245, 565, 1324, 3049, 7066, 16199, 37187, 84887, 193532, 439600, 996818, 2253941, 5086980, 11454778, 25746467, 57756522, 129342179, 289153474, 645399011, 1438308839, 3200671082, 7112360474, 15783402471, 34980122720, 77428353682

Offset: 0

Thomas Wieder, Jul 09 2005

nonn

Unlabeled analog of A097237.

Primes in this sequence include: a(3) = 3, a(11) = 3049, a(19) = 2253941, a(22) = 25746467. Semiprimes in this sequence include: a(9) = 565 = 5 * 113, a(12) = 7066 = 2 * 3533, a(13) = 16199 = 97 * 167, a(14) = 37187 = 41 * 907, a(15) = 84887 = 11 * 7717, a(18) = 996818 = 2 * 498409, a(24) = 129342179 = 23 * 5623573, a(30) = 15783402471 = 3 * 5261134157. - Jonathan Vos Post, Jul 10 2005

			Let * denote an unlabeled element.
Let : denote a delimiter between two levels of a hierarchy.
Let | denote a delimiter between two subhierarchies.
a(4) = 8 because we have *:*:*:*, ***:*, **:*:*, *:*|*:*, *:***, **:**, *:**:*, *:*:**.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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, A034691, A097237.

SetSeqXSetU := [S, {S=Set(U), U=Sequence(V,card>=2),V=Set(Z,card>=1)},unlabeled]; seq(count(SetSeqXSetU,size=j),j=0..30); #where x is an integer 1, 2, 3,... # x=2 gives 2 levels per society.

nmax = 40; CoefficientList[Series[E^Sum[x^(2*k)/(k*(1 - x^k)*(1 - 2*x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 08 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^(2^(k-1)-1). - Ilya Gutkovskiy, Jun 07 2018

a(n) ~ 2^n * exp(sqrt(2*n) - 5/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} 1/(k*(2^k-1)*(2^k-2)) = 0.0927294481510243482503144824759369647388... - Vaclav Kotesovec, Jun 08 2018

A136723 Number of preferential arrangements (or hierarchical orderings) on the connected graphs on n labeled nodes.

Original entry on oeis.org

1, 1, 3, 52, 2850, 393848, 125054832, 88260845008, 137304025714320, 469859118159233792, 3527181890877230433408, 57833314494643038031674112, 2060645597746315164145860149760, 158727775101107953869596632383822848, 26301662700662611321804753231934678909952

Offset: 0

Thomas Wieder, Jan 19 2008; corrected Jan 19 2008

Figure n3 demonstrates all 4*13=52 hierarchical orderings on n=3 connected points. In addition, the pink pictures describe the 10 cases where not all or no points are connected.

			There is A001187(2)=1 connected graph for n=2 labeled elements: The chain 1-2.
The chain gives us 3 hierarchical orderings:
1-2
1
|
2
2
|
1

Alois P. Heinz, Table of n, a(n) for n = 0..76
Thomas Wieder, Figure n3

Cf. A001187, A000670, A136722, A034691, A075729.

a(n) = A001187(n)*A000670(n);

Offset corrected by Alois P. Heinz, Dec 16 2014

A256549 Triangle read by rows, T(n,k) = {n,k}*h(k), where {n,k} are the Stirling set numbers and h(k) = hypergeom([-k+1,-k],[],1), for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 9, 13, 0, 1, 21, 78, 73, 0, 1, 45, 325, 730, 501, 0, 1, 93, 1170, 4745, 7515, 4051, 0, 1, 189, 3913, 25550, 70140, 85071, 37633, 0, 1, 381, 12558, 124173, 526050, 1077566, 1053724, 394353, 0, 1, 765, 39325, 567210, 3482451, 10718946, 17386446, 14196708, 4596553

Offset: 0

Peter Luschny, Apr 12 2015

			Triangle starts:
[1]
[0, 1]
[0, 1,  3]
[0, 1,  9,   13]
[0, 1, 21,   78,   73]
[0, 1, 45,  325,  730,  501]
[0, 1, 93, 1170, 4745, 7515, 4051]

Cf. A000110, A000262, A048993, A068156, A075729.

A000262 = lambda n: simplify(hypergeometric([-n+1, -n], [], 1))
A256549 = lambda n,k: A000262(k)*stirling_number2(n,k)
for n in range(7): [A256549(n,k) for k in (0..n)]

Row sums are A075729.
Alternating row sums are the signed Bell numbers (-1)^n*A000110(n).
T(n,k) = A048993(n,k)*A000262(k).
T(n,n) = A000262(n).
T(n+2,2) = A068156(n).

A355720 Expansion of e.g.f. exp( x/(2 - exp(x)) ).

Original entry on oeis.org

1, 1, 3, 16, 113, 986, 10237, 123096, 1680737, 25668766, 433329461, 8009178596, 160802065393, 3483842906610, 80992799730221, 2010720004254856, 53081510001375041, 1484613248976841958, 43846812123456425221, 1363477059263944382604

Offset: 0

Seiichi Manyama, Jul 15 2022

nonn

Cf. A000248, A355718, A355719.

Cf. A000670, A052882, A075729.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(2-exp(x)))))

a000670(n) = sum(k=0, n, k!*stirling(n, k, 2));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*a000670(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} A052882(k) * binomial(n-1,k-1) * a(n-k).

a(n) ~ n^(n - 1/4) * exp(sqrt(2*n) - 1/4 - n) / (sqrt(2) * log(2)^n). - Vaclav Kotesovec, Jul 15 2022

A075756 Number of hierarchies of hierarchies of hierarchies on n points.

Original entry on oeis.org

1, 1, 6, 52, 588, 8174, 134537, 2554647, 54909468, 1316675221, 34820961457, 1006230148609, 31529224324159, 1064355502971193, 38497326001639439, 1484865225798412485, 60822449267067095601, 2636248249383130776940, 120520100503562054999860, 5794815395039941996204424

Offset: 0

N. J. A. Sloane, Oct 15 2002

nonn

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, A075744.

m = 20;
f[x_] = E^(-1 + E^(-1 + 1/(2 - E^x)));
CoefficientList[Exp[f[x] - 1] + O[x]^m, x]*Range[0, m - 1]! (* Jean-François Alcover, Feb 24 2019 *)

E.g.f.: exp(f(x)-1) where f(x) = e.g.f. for A075744.

A104533 E.g.f.: exp(2x/(1-2x)).

Original entry on oeis.org

1, 2, 12, 104, 1168, 16032, 259264, 4817024, 100954368, 2353435136, 60355677184, 1687701792768, 51077784506368, 1662782678736896, 57917727119818752, 2148722382829027328, 84569896954751942656, 3518839711497761980416, 154306731918073225019392

Offset: 0

Thomas Wieder, Mar 13 2005

nonn

Number of hierarchical orderings for n labeled elements (see A075729) when there are two kinds A and B of elements.

			Let "a_i" and "b_j" be elements situated in the classes A and B with _i and _j as labels. Let : denote a separator among levels (ranks). Let | denote a separator among groups. E.g., a_1:b_2|b_1 is a hierarchy composed of two groups which contain three elements in total.
a(2) = 12 from b_2:b_1, b_2:a_1, b_2|b_1, a_1:a_2, b_2:a_1, a_1|a_2, a_1:b_2, a_2:a_1, b_1:a_2, a_2:b_1, b_1|a_2, b_2:b_1.

Seiichi Manyama, Table of n, a(n) for n = 0..398 (terms 0..150 from Alois P. Heinz)
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, A034691, A034899.

Equals 2^n * A000262(n).

SetSeqUnnL := [T, {T=Set(S,card>=1), S=Sequence(U,card>=1), U=Union(a,b),a=Atom, b=Atom},labeled]; seq(count(SetSeqUnnL,size=j),j=1..20);
A104533 := proc(n::integer) local i,j,prttnlst,prttn,liste,ZahlVerschiedenerTeile,H,Mltplztt; Mltplztt:=vector[1000]; prttnlst:=partition(n); H := 0; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; liste := convert(prttn,multiset); ZahlVerschiedenerTeile := nops(liste); for j from 1 to ZahlVerschiedenerTeile do Mltplztt[j] := op(2,op(j,liste)); od; H := H + (n!/mul(Mltplztt[j]!,j=1..ZahlVerschiedenerTeile)) * 2^n; od; print(n,H); end proc;

CoefficientList[Exp[2 x/(1 - 2 x)] + O[x]^21, x]*Range[0, 20]!
(* or: *)
a[0] = 1; a[n_] := 2^n*n!*Hypergeometric1F1[n + 1, 2, 1]/E;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 10 2017 *)

a(n) = 2^n*A000262(n) = 2^n*n!*Sum_{k=0..n} C(n-1,k)/(k+1)!. - Paul Barry, Apr 28 2007
With p(n) = the number of integer partitions of n, d(i) = the number of different parts of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} n!/(prod_{j=1}^{d(i)} m(i, j)!) * 2^(n)
E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)*(1-2*x)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 09 2013
E.g.f.: E(0) - 1, where E(k) = 2 + 2*x/((2*k+1)*(1-2*x) - 2*x/E(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Dec 31 2013

Edited by N. J. A. Sloane, May 06 2008, at the suggestion of Joerg Arndt

cp's OEIS Frontend

A354291 Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(4 - 3*exp(x)) = e.g.f. for A032033.

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A088729 Matrix product of Stirling2-triangle A008277(n,k) and unsigned Lah-triangle |A008297(n,k)|.

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A099880 Number of preferential arrangements (or simple hierarchies) of 2*n labeled elements with two kinds of elements (where each kind has n elements).

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A109509 Number of hierarchical orderings with at least 2 elements on each level for n unlabeled elements. Unlabeled analog of A097236.

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A110045 Number of hierarchical orderings ("societies") of n unlabeled elements ("individuals") with at least two occupied levels.

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A136723 Number of preferential arrangements (or hierarchical orderings) on the connected graphs on n labeled nodes.

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A256549 Triangle read by rows, T(n,k) = {n,k}*h(k), where {n,k} are the Stirling set numbers and h(k) = hypergeom([-k+1,-k],[],1), for n>=0 and 0<=k<=n.

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A355720 Expansion of e.g.f. exp( x/(2 - exp(x)) ).