A129506 -id:A129506 - cp's OEIS frontend

A386636 Triangle read by rows where T(n,k) is the number of inseparable type set partitions of {1..n} into k blocks.

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 1, 5, 0, 0, 0, 0, 1, 21, 15, 0, 0, 0, 0, 1, 28, 21, 0, 0, 0, 0, 0, 1, 92, 196, 56, 0, 0, 0, 0, 0, 1, 129, 288, 84, 0, 0, 0, 0, 0, 0, 1, 385, 1875, 1380, 210, 0, 0, 0, 0, 0, 0, 1, 561, 2860, 2145, 330, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 10 2025

A set partition is of inseparable type iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
A set partition is also of inseparable type iff its greatest block size is at least 2 more than the sum of all its other block sizes.
This is different from inseparable partitions (A325535) and partitions of inseparable type (A386638 or A025065).

			Row n = 6 counts the following set partitions:
  .  {123456}  {1}{23456}  {1}{2}{3456}  .  .  .
               {12}{3456}  {1}{2345}{6}
               {13}{2456}  {1}{2346}{5}
               {14}{2356}  {1}{2356}{4}
               {15}{2346}  {1}{2456}{3}
               {16}{2345}  {1234}{5}{6}
               {1234}{56}  {1235}{4}{6}
               {1235}{46}  {1236}{4}{5}
               {1236}{45}  {1245}{3}{6}
               {1245}{36}  {1246}{3}{5}
               {1246}{35}  {1256}{3}{4}
               {1256}{34}  {1345}{2}{6}
               {1345}{26}  {1346}{2}{5}
               {1346}{25}  {1356}{2}{4}
               {1356}{24}  {1456}{2}{3}
               {1456}{23}
               {12345}{6}
               {12346}{5}
               {12356}{4}
               {12456}{3}
               {13456}{2}
Triangle begins:
    0
    0    0
    0    1    0
    0    1    0    0
    0    1    4    0    0
    0    1    5    0    0    0
    0    1   21   15    0    0    0
    0    1   28   21    0    0    0    0
    0    1   92  196   56    0    0    0    0
    0    1  129  288   84    0    0    0    0    0
    0    1  385 1875 1380  210    0    0    0    0    0

For separable partitions we have A386583, sums A325534, ranks A335433.
For inseparable partitions we have A386584, sums A325535, ranks A335448.
For separable type partitions we have A386585, sums A336106, ranks A335127.
For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
Row sums are A386634.
The complement is counted by A386635, row sums A386633.
A000110 counts set partitions, row sums of A048993.
A000670 counts ordered set partitions.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A279790 counts disjoint families on strongly normal multisets.
A335434 counts separable factorizations, inseparable A333487.
A336103 counts normal separable multisets, inseparable A336102.
A386587 counts disjoint families of strict partitions of each prime exponent.
Cf. A008284, A072233, A097805, A106351, A116861, A124762, A129506, A190945, A239455, A335125, A374252, A386575.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
Table[Length[Select[sps[Range[n]],Length[#]==k&&stnseps[#]=={}&]],{n,0,5},{k,0,n}]

A106533 The rumor constant: decimal expansion of the number x defined by xe^(2 - 2x) = 1.

Original entry on oeis.org

2, 0, 3, 1, 8, 7, 8, 6, 9, 9, 7, 9, 9, 7, 9, 9, 5, 3, 8, 3, 8, 4, 7, 9, 0, 6, 2, 0, 6, 2, 4, 1, 9, 8, 7, 9, 1, 0, 5, 4, 9, 8, 7, 8, 7, 5, 9, 0, 5, 7, 0, 3, 1, 7, 5, 0, 0, 2, 4, 7, 7, 4, 4, 1, 5, 1, 9, 5, 7, 5, 0, 7, 5, 9, 1, 9, 0, 6, 0, 2, 4, 8, 8, 3, 6, 2, 5, 0, 3, 6, 1, 6, 9, 0, 7, 7, 9, 6, 4, 2, 9, 1, 4, 6, 9

Offset: 0

Robert G. Wilson v, May 03 2005

			c = 0.20318786997997995383847906206241987910549878759057031750024774...

G. C. Greubel, Table of n, a(n) for n = 0..5000
L. Bayon, P. Fortuny Ayuso, J. M. Grau, A. M. Oller-Marcen, M. M. Ruiz, The Best-or-Worst and the Postdoc problems, arXiv:1706.07185 [math.PR], 2017.
L. Bayon, J. Grau, A. M. Oller-Marcen, M. Ruiz, P. M. Suarez, A variant of the Secretary Problem: the Best or the Worst, arXiv preprint arXiv:1603.03928 [math.PR], 2016.
D. J. Daley and D. G. Kendall, Stochastic rumours, Journal of the Institute of Mathematics and Its Applications 1:42-55, 1965.
Steven R. Finch, Errata and Addenda to Mathematical Constants.
José María Grau Ribas, A New Proof and Extension of the Odds-Theorem, arXiv:1812.09255 [math.PR], 2018.
José María Grau Ribas, An extension of the Last-Success-Problem, Statistics & Probability Letters (2020) Vol. 156, 108591.
Brian Hayes, Rumours and Errours, American Scientist, Vol. 93, Nbr. 3, 2005, pp. 207-211.
Index entries for transcendental numbers.

Cf. A007456, A129506, A217900, A217910, A256500.

RealDigits[ -ProductLog[ -2/E^2]/2, 10, 111][[1]]
RealDigits[x/.FindRoot[x E^(2-2x)==1,{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Jul 05 2025 *)

solve(x=0, 0.5, x*exp(2-2*x)-1) \\ Michel Marcus, Mar 13 2016

Solution to x*exp(2 - 2*x) = 1 with x not equal to 1.
Equals -1/2*LambertW(-2*exp(-2)). - Vladeta Jovovic, May 30 2005
Constant c satisfies: exp(c*x)/(1-2*c) = Sum_{n>=0} (x + 2*n)^n * exp(-2*n)/n!. - Paul D. Hanna, Mar 12 2016
Equals (2-A256500)/2. - Miko Labalan, Dec 18 2024

A129505 Number of permutations of 2n-1 objects with exactly n cycles.

Original entry on oeis.org

1, 3, 35, 735, 22449, 902055, 44990231, 2681453775, 185953177553, 14710753408923, 1307535010540395, 129006659818331295, 13990945200239106865, 1654339178844590073615, 211821088794711294496815, 29197210605623737977801375, 4310704065427058593776844065

Offset: 1

Paul D. Hanna, Apr 18 2007

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..200
D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.

Cf. A008275, A129506.

Cf. A238685.

a129505 n = abs $ a008275 (2 * n - 1) n -- Reinhard Zumkeller, Mar 02 2014

t[n_] := SymmetricPolynomial[n, Range[1, 2 n]]
Table[t[n], {n, 1, 6}]  (* A129505 *)
(* Clark Kimberling, Dec 30 2011 *)
Table[Abs[StirlingS1[2*n-1, n]], {n, 1, 20}] (* Vaclav Kotesovec, Dec 28 2013 *)

a(n):=((2*n+1)*(-1)^n*((sum((stirling1(2*i-1,i)*binomial(2*n,2*i-1)* stirling1(2*(n-i)+1,n-i))/((n-i)*binomial(n,i)),i,1,n-1)) -n*stirling1(2*n-1,n) +stirling1(2*n,n)))/(n+1); /* Vladimir Kruchinin, Feb 28 2013 */

a(n):=coeff(expand(product(x+i,i,1,2*(n-1))),x,(n-1)); /* Lorraine Lee, Oct 12 2019 */

a(n)=polcoeff(prod(k=0,2*n-2,1+k*x),n-1)

vector(66, n, abs( stirling(2*n-1, n, 1) ) ) /* Joerg Arndt, Jun 09 2012 */

from sympy.functions.combinatorial.numbers import stirling
def A129505(n): return stirling((n<<1)-1,n,kind=1) # Chai Wah Wu, Jun 08 2025

Unsigned central Stirling numbers of the first kind:
G.f.: A(x) = Sum_{n>=0} a(n)*(2*n-1)!/n!*x^n = B'(x), where B(x) satisfies B(x)^2 = x*log(1/(1-B(x))). - Vladimir Kruchinin, Jun 10 2012
a(n) = ((2*n+1)*(-1)^n*((Sum_{i=1..n-1} (Stirling1(2*i-1,i)*C(2*n,2*i-1)*Stirling1(2*(n-i)+1,n-i))/((n-i)*C(n,i)))-n*Stirling1(2*n-1,n) + Stirling1(2*n,n)))/(n+1). - Vladimir Kruchinin, Feb 28 2013
a(n) ~ (1+2*c)/(8*c*sqrt(Pi*(-1-c))) * (-8*c^2/(exp(1)*(1+2*c)))^n * n^(n-3/2), where c = LambertW(-1,-1/(2*exp(1/2))). - Vaclav Kotesovec, Dec 28 2013
a(n) = abs(C(2*n-1,n-1)*Sum_{i=1..n-1} (Stirling1(n-1,n-i-1)*Stirling1(n,i+1)/C(n-1,i))). - Chai Wah Wu, Jun 08 2025

Minor edits by Vaclav Kotesovec, Mar 31 2014

A233734 Central terms of triangles A019538 and A090582.

Original entry on oeis.org

1, 6, 150, 8400, 834120, 129230640, 28805736960, 8734434508800, 3457819037312640, 1732015476199008000, 1070842073499515116800, 800968643959240044288000, 712900933001021056900608000, 744602794912654938776355840000, 901893717412811100821094451200000

Offset: 1

Reinhard Zumkeller, Dec 15 2013

nonn

a(n) = A019538(2*n-1,n) = A090582(2*n-1,n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..200

Haskell
```
a233734 n = a019538 (2 * n - 1) n
```

a(n) = A129506(n) * n!.

A258170 T(n,k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * A185651(n,i); triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 8, 6, 1, 0, 5, 15, 25, 10, 1, 0, 6, 36, 91, 65, 15, 1, 0, 7, 63, 301, 350, 140, 21, 1, 0, 8, 136, 972, 1702, 1050, 266, 28, 1, 0, 9, 261, 3027, 7770, 6951, 2646, 462, 36, 1, 0, 10, 530, 9355, 34115, 42526, 22827, 5880, 750, 45, 1

Offset: 0

Alois P. Heinz, May 22 2015

			Triangle T(n,k) begins:
  0;
  0,  1;
  0,  2,   1;
  0,  3,   3,    1;
  0,  4,   8,    6,     1;
  0,  5,  15,   25,    10,     1;
  0,  6,  36,   91,    65,    15,     1;
  0,  7,  63,  301,   350,   140,    21,    1;
  0,  8, 136,  972,  1702,  1050,   266,   28,   1;
  0,  9, 261, 3027,  7770,  6951,  2646,  462,  36,  1;
  0, 10, 530, 9355, 34115, 42526, 22827, 5880, 750, 45, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-1 give: A000004, A000027.
Row sums give A258171.
Main diagonal gives A057427.
T(2*n+1,n+1) gives A129506(n+1).
Cf. A008277, A185651, A327029.

with(numtheory):
A:= proc(n, k) option remember;
      add(phi(d)*k^(n/d), d=divisors(n))
    end:
T:= (n, k)-> add((-1)^(k-i)*binomial(k, i)*A(n, i), i=0..k)/k!:
seq(seq(T(n, k), k=0..n), n=0..12);

A[n_, k_] := A[n, k] = DivisorSum[n, EulerPhi[#]*k^(n/#)&];
T[n_, k_] := Sum[(-1)^(k-i)*Binomial[k, i]*A[n, i], {i, 0, k}]/k!; T[0, 0] = 0;
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 25 2017, translated from Maple *)

# uses[DivisorTriangle from A327029]
DivisorTriangle(euler_phi, stirling_number2, 10) # Peter Luschny, Aug 24 2019

T(n,k) = 1/k! * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * A185651(n,i).
From Petros Hadjicostas, Sep 07 2018: (Start)
Conjecture 1: T(n,k) = Stirling2(n,k) for k >= 1 and k <= n <= 2*k - 1.
Conjecture 2: T(n,k) = Stirling2(n,k) for k >= 2 and n prime >= 2.
Here, Stirling2(n,k) = A008277(n,k).
(End)

A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets.

Original entry on oeis.org

1, 15, 966, 145750, 40075035, 17505749898, 11143554045652, 9741955019900400, 11201516780955125625, 16392038075086211019625, 29749840488672593296243236, 65580126734167548918100615020, 172597131674172062132363512309613, 534584200037719212882636004559739000, 1924887533450780657560944228447179522880, 7973126100358260458973226689851075932667520, 37645241791600906804871080818625037726247519045

Offset: 1

Paul D. Hanna, Jul 03 2016

nonn

a(n) is divisible by the triangular numbers: a(n) / (n*(n+1)/2) = A274712(n).

			O.g.f.: A(x) = x + 15*x^2 + 966*x^3 + 145750*x^4 + 40075035*x^5 + 17505749898*x^6 + 11143554045652*x^7 + 9741955019900400*x^8 +...
where
A(x) = exp(-x)*x + 2^5*exp(-2^3*x)*x^2/2! + 3^8*exp(-3^3*x)*x^3/3! + 4^11*exp(-4^3*x)*x^4/4! + 5^14*exp(-5^3*x)*x^5/5! + 6^17*exp(-6^3*x)*x^6/6! + 7^20*exp(-7^3*x)*x^7/7! + 8^23*exp(-8^3*x)*x^8/8! +...+ n^(3*n-1)*exp(-n^3*x)*x^n/n! +...
simplifies to an integer series.

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A129506, A217913, A274712, A008277.

Table[StirlingS2[3*n - 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Jul 06 2016 *)

{a(n) = abs( stirling(3*n-1, n, 2) )}
for(n=1, 20, print1(a(n), ", "))

{a(n) = 1/n! * sum(k=0, n, (-1)^(n-k) * binomial(n, k) * k^(3*n-1))}
for(n=1, 20, print1(a(n), ", "))

{a(n) = polcoeff( 1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), 2*n-1)}
for(n=1, 20, print1(a(n), ", "))

{a(n) = polcoeff( sum(m=1, n, m^(3*m-1) * x^m * exp(-m^3*x +x*O(x^n))/m!), n)}
for(n=1, 20, print1(a(n), ", "))

O.g.f.: Sum_{n>=1} n^(3*n-1) * exp(-n^3*x) * x^n / n!, an integer series.
a(n) = A008277(3*n-1,n) for n>=1, where A008277 are the Stirling numbers of the second kind.
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(3*n-1).
a(n) = [x^(2*n-1)] 1 / Product_{k=1..n} (1 - k*x).
a(n) ~ 3^(3*n-1) * n^(2*n-3/2) / (exp(2*n) * c^n * (3-c)^(2*n-1) * sqrt(2*Pi*(1-c))), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968... = -A226750. - Vaclav Kotesovec, Jul 06 2016

A383869 a(n) = [x^n] 1/Product_{k=0..n} (1 - (n+k)*x).

Original entry on oeis.org

1, 3, 55, 1890, 95781, 6427575, 537306484, 53791898160, 6275077781973, 835898091070185, 125195263380478655, 20825548503275385870, 3809430011164368694260, 759987002381075483922180, 164221938436980055710082200, 38209754165858724861944820000, 9524153723280871205135022364485

Offset: 0

Seiichi Manyama, May 13 2025

nonn

Central terms of triangle A143395.

Cf. A007820, A129506.

Table[SeriesCoefficient[1/Product[(1 - (n + k)*x), {k, 0, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 17 2025 *)

a(n) = sum(k=0, n, (-1)^(n-k)*(n+k)^(2*n)*binomial(n, k))/n!;

a(n) = (1/n!) * Sum_{k=0..n} (-1)^(n-k) * (n+k)^(2*n) * binomial(n,k).
a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*n,n+k) * Stirling2(n+k,n).
a(n) = Sum_{k=0..n} (-1)^k * (2*n)^(n-k) * binomial(2*n,n+k) * Stirling2(n+k,n).
a(n) ~ (r-1)^((r-1)*n) * (1+r)^(2*n + 1) * exp(n) * n^(n - 1/2) / (sqrt(2*Pi*(1 + (4-r)*r)) * r^(r*n)), where r = 2.106565648173949260853515992430777519716829316322... is the root of the equation exp(2/(1+r)) = r/(r-1). - Vaclav Kotesovec, May 17 2025

A343278 a(n) = Stirling2(n, ceiling(n/2)).

Original entry on oeis.org

1, 1, 1, 3, 7, 25, 90, 350, 1701, 6951, 42525, 179487, 1323652, 5715424, 49329280, 216627840, 2141764053, 9528822303, 106175395755, 477297033785, 5917584964655, 26826851689001, 366282500870286, 1672162773483930, 24930204590758260, 114485073343744260

Offset: 0

Peter Luschny, Apr 20 2021

nonn

Number of partitions of an n-set into ceiling(n/2) nonempty subsets.

Bisection gives A007820 (even part),

Cf. A048993, A129506, A343279.

Table[StirlingS2[n, Ceiling[n/2]], {n, 0, 25}] (* Amiram Eldar, Apr 20 2021 *)

a(n) = stirling(n, ceil(n/2), 2); \\ Michel Marcus, Apr 20 2021

A343279 a(n) = Stirling2(n, floor(n/2)).

Original entry on oeis.org

1, 0, 1, 1, 7, 15, 90, 301, 1701, 7770, 42525, 246730, 1323652, 9321312, 49329280, 408741333, 2141764053, 20415995028, 106175395755, 1144614626805, 5917584964655, 71187132291275, 366282500870286, 4864251308951100, 24930204590758260, 362262620784874680

Offset: 0

Peter Luschny, Apr 20 2021

nonn

Number of partitions of an n-set into floor(n/2) nonempty subsets.

Cf. A007820, A048993, A129506, A343278.

Table[StirlingS2[n, Floor[n/2]], {n, 0, 25}] (* Amiram Eldar, Apr 20 2021 *)

a(n) = stirling(n, n\2, 2); \\ Michel Marcus, Apr 20 2021

A226703 Triangle read by rows: T(n,k) = binomial(2n,k)Stirling2(2*n-k,n).

Original entry on oeis.org

1, 1, 2, 7, 12, 6, 90, 150, 90, 20, 1701, 2800, 1820, 560, 70, 42525, 69510, 47250, 16800, 3150, 252, 1323652, 2153844, 1506582, 582120, 131670, 16632, 924, 49329280, 80015936, 57093036, 23291268, 5885880, 924924, 84084, 3432, 2141764053, 3466045440, 2509478400, 1063782720, 289429140, 51891840, 6006000, 411840, 12870

Offset: 0

Vladimir Kruchinin, Jun 15 2013

Polynomials based on Extended Tepper's Identity
P(n,x)=sum(j=0..n, (-1)^(n-j)*binomial(n,j)*(x+j)^(2*n))/n!.
P(n,x)=sum(j=0..n, binomial(2*n,j)*stirling2(2*n-j,n)*x^j).
P(n,1)=A129506(n).

			1,
1 +2*x,
7 +12*x +6*x^2,
90 +150*x +90*x^2 +20*x^3,
1701 +2800*x +1820*x^2 +560*x^3 +70*x^4.

G. P. Egorychev. “Integral Representation and the Computation of Combinatorial Sums.” Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, (1984).
F. J. Papp. “Another Proof of Tepper’s Inequality.” Math. Magazine 45 (1972): 119-121.

Cf. A000984, A007820, A129506.

Flatten[Table[Binomial[2n,k]StirlingS2[2n-k,n],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 19 2013 *)

T(n,k) = binomial(2*n,k)*stirling2(2*n-k,n).
T(n,n) = A000984(n).
T(n,0) = A007820(n).

cp's OEIS Frontend

A386636 Triangle read by rows where T(n,k) is the number of inseparable type set partitions of {1..n} into k blocks.

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A106533 The rumor constant: decimal expansion of the number x defined by x*e^(2 - 2*x) = 1.

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A129505 Number of permutations of 2n-1 objects with exactly n cycles.

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A233734 Central terms of triangles A019538 and A090582.

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A258170 T(n,k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i) * C(k,i) * A185651(n,i); triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

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A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets.

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A383869 a(n) = [x^n] 1/Product_{k=0..n} (1 - (n+k)*x).

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A106533 The rumor constant: decimal expansion of the number x defined by xe^(2 - 2x) = 1.

A226703 Triangle read by rows: T(n,k) = binomial(2n,k)Stirling2(2*n-k,n).