A083355 -id:A083355 - cp's OEIS frontend

A232598 T(n,k) = Stirling2(n,k) * OrderedBell(k).

1, 1, 3, 1, 9, 13, 1, 21, 78, 75, 1, 45, 325, 750, 541, 1, 93, 1170, 4875, 8115, 4683, 1, 189, 3913, 26250, 75740, 98343, 47293, 1, 381, 12558, 127575, 568050, 1245678, 1324204, 545835, 1, 765, 39325, 582750, 3760491, 12391218, 21849366, 19650060, 7087261

Offset: 1

Tilman Piesk, Nov 26 2013

T(n,k) is the number of preferential arrangements of the k-part partitions of the set {1...n}.
2*T(n,k) is the number of formulas in first order logic that have an n-place predicate and use k variables, but don't include a negator.
4*T(n,k) is the number of such formulas that may include an negator.
The entries T(n,n) are A000670(n), i.e. the ordered Bell numbers.

			Let the colon ":" be a separator between two levels. E.g. in {1,2}:{3} the set {1,2} is on the first level, the set {3} is on the second level.
Compare descriptions of A083355 and A233357.
a(3,1) = 1:
{1,2,3}
a(3,2) = 9:
{1,2}{3}
{1,3}{2}
{2,3}{1}
{1,2}:{3}   {3}:{1,2}
{1,3}:{2}   {2}:{1,3}
{2,3}:{1}   {1}:{2,3}
a(3,3) = 13:
{1}{2}{3}
{1}{2}:{3}   {3}:{1}{2}
{1}{3}:{2}   {2}:{1}{3}
{2}{3}:{1}   {1}:{2}{3}
{1}:{2}:{3}
{1}:{3}:{2}
{2}:{1}:{3}
{2}:{3}:{1}
{3}:{1}:{2}
{3}:{2}:{1}
Triangle begins:
     k = 1   2     3      4      5       6       7      8          sums
n
1        1                                                            1
2        1   3                                                        4
3        1   9    13                                                 23
4        1  21    78     75                                         175
5        1  45   325    750    541                                 1662
6        1  93  1170   4875   8115    4683                        18937
7        1 189  3913  26250  75740   98343   47293               251729
8        1 381 12558 127575 568050 1245678 1324204 545835       3824282

Tilman Piesk, First 100 rows, flattened
Tilman Piesk, Preferential arrangements of set partitions (Wikiversity)

A008277 (Stirling2), A000670 (ordered Bell), A068156 (column k=2), A083355 (row sums: number of preferential arrangements), A233357 (number of preferential arrangements by number of levels).

T(n,k) = A008277(n,k) * A000670(k).
T(n,n) = A000670(n).
T(n,2) = A068156(n-1).
From Peter Bala, Nov 27 2013: (Start)
E.g.f.: 1/( 2 - exp(x*(exp(t) - 1)) ) = 1 + x*t + (x + 3*x^2)*t^2/2! + (x + 9*x^2 + 13*x^3)*t^3/3! + ....
Recurrence equation (for entries not on main diagonal): (n - k)*T(n,k) = C(n,1)*T(n-1,k) - C(n,2)*T(n-2,k) + C(n,3)*T(n-3,k) - ... (End)

A300696 a(n) is the number of n-place formulas in first-order logic when variables are allowed to coincide.

Original entry on oeis.org

1, 2, 8, 46, 350, 3324, 37874, 503458, 7648564, 130722474, 2482437926, 51856030736, 1181704007894, 29172943488602, 775597634145192, 22093062633006326, 671280598744505190, 21671112459225274300, 740767465663838556074, 26727829360555847269034

Offset: 0

Tilman Piesk, Mar 13 2018

nonn

An example of a 3-place formula in predicate logic is Ex Ay Ez P(x,y,z). The number of different formulas when x, y, z have to be different is A000629(3) = 26. When variables are allowed to coincide that means that there are 20 more formulas like, e.g., Ex Ay P(x,x,y) or Ex P(x,x,x).
a(n) is the number of vertices in a cocoon concertina n-cube and the sum of row n in A300695, which shows the number of vertices in that structure by rank. A000629(n) by comparison is the number of vertices in the convex concertina n-cube.
The differences with A000629, i.e., the numbers of formulas with coinciding variables, are 0, 0, 2, 20, 200, 2242, 28508, 408872, 6556894, 116547952, 2277942800, ...

Tilman Piesk, Table of n, a(n) for n = 0..150
Tilman Piesk, List of all 46 formulas with 3 places
Tilman Piesk, Python code used to generate the sequence

Cf. A000629, A083355, A300695.

a(0) = 1, a(n) = 2 * A083355(n) for n > 0.

A099391 Expansion of e.g.f. 1/(2 - exp(exp(exp(x) - 1) - 1)).

Original entry on oeis.org

1, 1, 5, 36, 342, 4048, 57437, 950512, 17975438, 382424397, 9039989107, 235062317196, 6667866337309, 204905200542916, 6781157167505291, 240446179599065951, 9094120016963808935, 365453749501228063845

Offset: 0

Ralf Stephan, Oct 18 2004

nonn

K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem [J. Phys. A 37 (2004), 3475-3487]

Column k=3 of A363007.
Row p=3 of A153278 (for n>=1).
Cf. A000110, A083355, A130410.

With[{nn=20},CoefficientList[Series[1/(2-Exp[Exp[Exp[x]-1]-1]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 10 2014 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(exp(exp(x)-1)-1)))) \\ Seiichi Manyama, May 12 2023

(1/2) Sum[k=0..inf, k^n/k! * Sum[r=1..inf, e^(-r)r^k/r!*Li(-r, 1/2e) ]], with Li the polylogarithm.
a(n) ~ n! / (2 * (1 + log(2)) * (1 + log(1 + log(2))) * log(1 + log(1 + log(2)))^(n+1)). - Vaclav Kotesovec, Jun 26 2022
a(n) = Sum_{k=0..n} Stirling2(n,k) * A083355(k). - Seiichi Manyama, May 12 2023

Definition clarified by Harvey P. Dale, Apr 10 2014

A233357 Triangle read by rows: T(n,k) = ((Stirling2)^2)(n,k) * k!

Original entry on oeis.org

1, 2, 2, 5, 12, 6, 15, 64, 72, 24, 52, 350, 660, 480, 120, 203, 2024, 5670, 6720, 3600, 720, 877, 12460, 48552, 83160, 71400, 30240, 5040, 4140, 81638, 424536, 983808, 1201200, 806400, 282240, 40320

Offset: 1

Tilman Piesk, Dec 07 2013

T(n,k) is the number of preferential arrangements with k levels of partitions of the set {1...n}.
2*T(n,k) is the number of formulas in first order logic that have an n-place predicate and k runs of A's and E's (universal and existential quantifiers, compare runs of 0's ans 1's counted by A005811), but don't include a negator.
4*T(n,k) is the number of such formulas that may include an negator.
T(n,k) is the number of partitions of an n-set into colored blocks, such that exactly k colors are used. T(3,2) = 12: 1a|23b, 1b|23a, 13a|2b, 13b|2a, 12a|3b, 12b|3a, 1a|2a|3b, 1b|2b|3a, 1a|2b|3a, 1b|2a|3b, 1a|2b|3b, 1b|2a|3a. - Alois P. Heinz, Sep 01 2019

			Let the colon ":" be a separator between two levels. E.g. in {1,2}:{3} the set {1,2} is on the first level, the set {3} is on the second level.
Compare descriptions of A083355 and A232598.
a(3,1)=5:
{1,2,3}
{1,2}{3}
{1,3}{2}
{2,3}{1}
{1}{2}{3}
a(3,2)=12:
{1,2}:{3}   {3}:{1,2}
{1,3}:{2}   {2}:{1,3}
{2,3}:{1}   {1}:{2,3}
{1}{2}:{3}   {3}:{1}{2}
{1}{3}:{2}   {2}:{1}{3}
{2}{3}:{1}   {1}:{2}{3}
a(3,3)=6:
{1}:{2}:{3}
{1}:{3}:{2}
{2}:{1}:{3}
{2}:{3}:{1}
{3}:{1}:{2}
{3}:{2}:{1}
Triangle begins:
       k = 1     2      3      4       5      6      7     8           sums
1          1                                                              1
2          2     2                                                        4
3          5    12      6                                                23
4         15    64     72     24                                        175
5         52   350    660    480     120                               1662
6        203  2024   5670   6720    3600    720                       18937
7        877 12460  48552  83160   71400  30240   5040               251729
8       4140 81638 424536 983808 1201200 806400 282240 40320        3824282

Tilman Piesk, First 100 rows, flattened
Tilman Piesk, Preferential arrangements of set partitions (Wikiversity)

A008277 (Stirling2), A039810 (square of Stirling2), A000110 (Bell), A000142 (factorials), A083355 (row sums: number of preferential arrangements), A232598 (number of preferential arrangements by number of blocks).

Cf. A130191.

S2 = A008277 (Stirling numbers of the second kind).
(S2)^2 = A039810 (matrix square of S2).
T(n,k) = ((S2)^2)(n,k) * k! = Sum(k<=i<=n) [ S2(n,i) * S2(i,k) ] * k!.
T(n,1) = Bell(n) = A000110(n).
T(n,2) = A052896(n).
T(n,n) = n! = A000142(n).
T(n,n-1) = n!*(n-1) = A062119(n).

A346433 E.g.f.: 1 / (2 - exp(2*(exp(x) - 1))).

Original entry on oeis.org

1, 2, 14, 142, 1910, 32094, 647126, 15223198, 409276054, 12378827166, 416006542550, 15378483225758, 620176642174742, 27094392220198814, 1274759052849262422, 64259896197635471006, 3455259407744574799254, 197401403111903906001310, 11941074177046918285056470

Offset: 0

Ilya Gutkovskiy, Jul 17 2021

nonn

Cf. A000670, A001861, A052555, A080253, A083355, A216794, A346417, A346432.

nmax = 18; CoefficientList[Series[1/(2 - Exp[2 (Exp[x]- 1)]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
Table[Sum[StirlingS2[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]

my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - exp(2*(exp(x) - 1))))) \\ Michel Marcus, Jul 18 2021

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A001861(k) * a(n-k).
a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^k * A000670(k).
a(n) ~ n! / (2*(2+log(2)) * (log(1+log(2)/2))^(n+1)). - Vaclav Kotesovec, Jul 27 2021

A328488 Expansion of e.g.f. 1 / (2 - exp(x * exp(x))).

Original entry on oeis.org

1, 1, 5, 34, 307, 3456, 46659, 734882, 13227995, 267871036, 6027206803, 149176155030, 4027831914099, 117816299188472, 3711283196035523, 125258162280991858, 4509378597919760779, 172486973301491042964, 6985853719202139488211, 298650859698906574479278

Offset: 0

Ilya Gutkovskiy, Oct 16 2019

nonn

Cf. A000248, A000670, A003725, A007550, A083355, A292952.

nmax = 19; CoefficientList[Series[1/(2 - Exp[x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000248(k) * a(n-k).

a(n) ~ n! / (2*log(2) * (1 + LambertW(log(2))) * LambertW(log(2))^n). - Vaclav Kotesovec, Oct 17 2019

A332254 E.g.f.: 1 / (2 - exp(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 1, 1, 10, 31, 271, 1534, 14603, 120173, 1310224, 13947517, 175477699, 2265702388, 32673218085, 492565328493, 8053045395018, 138334722101571, 2535114408394699, 48790865853110950, 991843960201311455, 21121971129683138297, 471959969940724275432

Offset: 0

Ilya Gutkovskiy, Feb 08 2020

nonn

Cf. A000296, A028248, A083355, A293037.

nmax = 22; CoefficientList[Series[1/(2 - Exp[Exp[x] - 1 - x]), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(1/(2 - exp(exp(x + O(x*x^n)) - 1 - x))))} \\ Andrew Howroyd, Feb 08 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000296(k) * a(n-k).

a(n) ~ n! * exp(1 - exp(c-1)/2) / ((1 - 2*exp(1-c)) * (c - 1 - log(2))^(n+1)), where c = -LambertW(-1, -exp(-1)/2) = 2.678346990016660653412884512094523... - Vaclav Kotesovec, Feb 08 2020

A332256 E.g.f.: 1 / (2 - exp(sinh(x))).

Original entry on oeis.org

1, 1, 3, 14, 87, 672, 6231, 67412, 833475, 11593140, 179170947, 3045978388, 56490392943, 1134970258372, 24557211519951, 569294311105300, 14077429483372251, 369861235318338404, 10289111057247180411, 302132879478864660340, 9338874072977661538119

Offset: 0

Ilya Gutkovskiy, Feb 08 2020

nonn

Cf. A000110, A000670, A003724, A006154, A053488, A083355.

nmax = 20; CoefficientList[Series[1/(2 - Exp[Sinh[x]]), {x, 0, nmax}], x] Range[0, nmax]!

seq(n)={Vec(serlaplace(1/(2 - exp(sinh(x + O(x*x^n))))))} \\ Andrew Howroyd, Feb 08 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A003724(k) * a(n-k).

a(n) ~ n! / (2 * sqrt(1 + log(2)^2) * (log(log(2) + sqrt(1 + log(2)^2)))^(n+1)). - Vaclav Kotesovec, Feb 08 2020

A335849 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Bell(k-1) * a(n-k).

Original entry on oeis.org

1, 1, 3, 14, 87, 675, 6282, 68201, 846183, 11811048, 183176577, 3124958179, 58157682072, 1172551946395, 25459025908899, 592263131497942, 14696581853565723, 387477880784385143, 10816856730117090114, 318739828787430822853, 9886623306152849028771

Offset: 0

Ilya Gutkovskiy, Jun 26 2020

nonn

Cf. A000110, A004122, A083355, A129247, A137551.

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] BellB[k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Exp[1]/(Exp[1] + ExpIntegralEi[1] - ExpIntegralEi[Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(1) / (exp(1) + Ei(1) - Ei(exp(x))), where Ei() is the exponential integral.

cp's OEIS Frontend

A232598 T(n,k) = Stirling2(n,k) * OrderedBell(k).

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A300696 a(n) is the number of n-place formulas in first-order logic when variables are allowed to coincide.

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A099391 Expansion of e.g.f. 1/(2 - exp(exp(exp(x) - 1) - 1)).

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A233357 Triangle read by rows: T(n,k) = ((Stirling2)^2)(n,k) * k!

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A346433 E.g.f.: 1 / (2 - exp(2*(exp(x) - 1))).

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Mathematica

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A328488 Expansion of e.g.f. 1 / (2 - exp(x * exp(x))).

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Mathematica

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A332254 E.g.f.: 1 / (2 - exp(exp(x) - 1 - x)).

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Mathematica

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A332256 E.g.f.: 1 / (2 - exp(sinh(x))).

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Mathematica

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A335849 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Bell(k-1) * a(n-k).

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