A000296 -id:A000296 - cp's OEIS frontend

A346740 Expansion of e.g.f.: exp(exp(x) - 5*x - 1).

1, -4, 17, -75, 340, -1573, 7393, -35178, 169035, -818603, 3989250, -19538555, 96084397, -474052868, 2344993157, -11624422855, 57722000172, -287012948441, 1428705217949, -7118044107698, 35489117143047, -177036294035559, 883588566571138, -4411213326568599, 22032317835916969

Offset: 0

Ilya Gutkovskiy, Jul 31 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..581

Cf. A000110, A000296, A126617, A196834, A196835, A290219, A346738, A346739.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(Exp(x) -5*x -1) ))) // G. C. Greubel, Jun 12 2024

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 5 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-5)^(n - k) BellB[k], {k, 0, n}], {n, 0, 24}]
a[0] = 1; a[n_] := a[n] = -5 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]

[factorial(n)*( exp(exp(x) -5*x -1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 5*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-5)^(n-k) * Bell(k).
a(n) = exp(-1) * Sum_{k>=0} (k - 5)^n / k!.
a(0) = 1; a(n) = -5 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).

A102286 Total number of odd blocks in all partitions of n-set.

Original entry on oeis.org

1, 2, 7, 24, 96, 418, 1989, 10216, 56275, 330424, 2057672, 13532060, 93633021, 679473694, 5156626991, 40824399712, 336406367196, 2879570703510, 25557841113625, 234822774979908, 2230107923204443, 21861817965483016, 220940261740238140, 2299258336094622008

Offset: 1

Vladeta Jovovic, Feb 19 2005

a(n) is also the number of set partitions of {1,2,...,n+1} in which the element 1 is in an even size block. - Geoffrey Critzer, Apr 02 2013

			a(3)=7 because we have (123), (1)/23, 12/(3), 13/(2), (1)/(2)/(3); the odd blocks are shown between parentheses.

Alois P. Heinz, Table of n, a(n) for n = 1..575

Cf. A000110, A000296, A005493.

G:=sinh(x)*exp(exp(x)-1): Gser:=series(G,x=0,30): seq(n!*coeff(Gser,x^n),n=1..25); # Emeric Deutsch
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
       add((p->(p+[0, `if`(i::odd, j, 0)*p[1]]))(
       b(n-i*j, i-1))*multinomial(n, n-i*j, i$j)/j!, j=0..n/i))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 16 2015

Range[0, nn]! CoefficientList[
  D[Series[Exp[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], y] /. y -> 1, x] (* Geoffrey Critzer, Aug 28 2012 *)
With[{nn=30},CoefficientList[Series[Sinh[x]Exp[Exp[x]-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 03 2021 *)

E.g.f: sinh(x)*exp(exp(x)-1).

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * Bell(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022

More terms from Emeric Deutsch, Mar 04 2005

A261137 Number of set partitions B'_t(n) of {1,2,...,t} into at most n parts, so that no part contains both 1 and t, or both i and i+1 with 1 <= i < t; triangle B'_t(n), t>=0, 0<=n<=t, read by rows.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 3, 4, 0, 0, 0, 5, 10, 11, 0, 0, 1, 11, 31, 40, 41, 0, 0, 0, 21, 91, 147, 161, 162, 0, 0, 1, 43, 274, 568, 694, 714, 715, 0, 0, 0, 85, 820, 2227, 3151, 3397, 3424, 3425, 0, 0, 1, 171, 2461, 8824, 14851, 17251, 17686, 17721, 17722

Offset: 0

Mark Wildon, Aug 10 2015

B'_t(n) is the number of sequences of t non-identity top-to-random shuffles that leave a deck of n cards invariant.
B't(n) = <chi^t, 1{Sym_n}> where chi is the degree n-1 constituent of the natural permutation character of the symmetric group Sym_n. This gives a combinatorial interpretation of B'_t(n) using sequences of box moves on Young diagrams.
B'_t(t) is the number of set partitions of a set of size t into parts of size at least 2 (A000296); this is also the number of cyclically spaced partitions of a set of size t.
B'_t(n) = B'_t(t) if n > t.

			Triangle starts:
  1;
  0, 0;
  0, 0, 1;
  0, 0, 0,  1;
  0, 0, 1,  3,   4;
  0, 0, 0,  5,  10,   11;
  0, 0, 1, 11,  31,   40,   41;
  0, 0, 0, 21,  91,  147,  161,  162;
  0, 0, 1, 43, 274,  568,  694,  714,  715;
  0, 0, 0, 85, 820, 2227, 3151, 3397, 3424, 3425;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D, arXiv:1507.04803 [math.CO], 2015.
D. E. Knuth and O. P. Lossers, Partitions of a circular set, Problem 11151 in Amer. Math. Monthly 114 (3), (2007), p 265, E_4.

Cf. A000296, A261139.

For columns n=3-8 see: A001045, A006342, A214142, A214167, A214188, A214239.

g:= proc(t, l, h) option remember; `if`(t=0, `if`(l=1, 0, x^h),
       add(`if`(j=l, 0, g(t-1, j, max(h,j))), j=1..h+1))
    end:
B:= t-> (p-> seq(add(coeff(p, x, j), j=0..i), i=0..t))(g(t, 0$2)):
seq(B(t), t=0..12);  # Alois P. Heinz, Aug 10 2015

StirPrimedGF[0, x_] := 1; StirPrimedGF[1, x_] := 0;
StirPrimedGF[n_, x_] := x^n/(1 + x)*Product[1/(1 - j*x), {j, 1, n - 1}];
StirPrimed[0, 0] := 1; StirPrimed[0, _] := 0;
StirPrimed[t_, n_] := Coefficient[Series[StirPrimedGF[n, x], {x, 0, t}], x^t];
BPrimed[t_, n_] := Sum[StirPrimed[t, m], {m, 0, n}]
(* Second program: *)
g[t_, l_, h_] := g[t, l, h] = If[t == 0, If[l == 1, 0, x^h], Sum[If[j == l, 0, g[t - 1, j, Max[h, j]]], {j, 1, h + 1}]];
B[t_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, t}] ][g[t, 0, 0]];
Table[B[t], {t, 0, 12}] // Flatten (* Jean-François Alcover, May 20 2016, after Alois P. Heinz *)

B't(n) = Sum{i=0..n} A261139(t,i).

A306357 Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.

Original entry on oeis.org

0, 1, 3, 6, 10, 20, 38, 70, 130, 240, 442, 814, 1498, 2756, 5070, 9326, 17154, 31552, 58034, 106742, 196330, 361108, 664182, 1221622, 2246914, 4132720, 7601258, 13980894, 25714874, 47297028, 86992798, 160004702, 294294530, 541292032, 995591266, 1831177830

Offset: 0

Gus Wiseman, Feb 10 2019

nonn

Cyclically successive means 1 is a successor of n.

Set partitions using these subsets are counted by A323949.

			The a(1) = 1 through a(5) = 20 stable subsets:
  {1}  {1}    {1}    {1}    {1}
       {2}    {2}    {2}    {2}
       {1,2}  {3}    {3}    {3}
              {1,2}  {4}    {4}
              {1,3}  {1,2}  {5}
              {2,3}  {1,3}  {1,2}
                     {1,4}  {1,3}
                     {2,3}  {1,4}
                     {2,4}  {1,5}
                     {3,4}  {2,3}
                            {2,4}
                            {2,5}
                            {3,4}
                            {3,5}
                            {4,5}
                            {1,2,4}
                            {1,3,4}
                            {1,3,5}
                            {2,3,5}
                            {2,4,5}

Cf. A000126, A000296, A001610, A001644, A169985, A306357, A323949, A323952, A323955.

stabsubs[g_]:=Select[Rest[Subsets[Union@@g]],Select[g,Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&];
Table[Length[stabsubs[Partition[Range[n],3,1,1]]],{n,15}]

For n >= 3 we have a(n) = A001644(n) - 1.
From Chai Wah Wu, Jan 06 2020: (Start)
a(n) = 2*a(n-1) - a(n-4) for n > 6.
G.f.: x*(x^5 + x^4 - 2*x^3 + x + 1)/(x^4 - 2*x + 1). (End)

A324014 Number of self-complementary set partitions of {1, ..., n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 9, 16, 43, 89, 250, 571, 1639

Offset: 0

Gus Wiseman, Feb 12 2019

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}.

			The  a(3) = 1 through a(6) = 9 self-complementary set partitions with no cyclical adjacencies:
  {{1}{2}{3}}  {{13}{24}}      {{14}{25}{3}}      {{135}{246}}
               {{1}{2}{3}{4}}  {{1}{24}{3}{5}}    {{13}{25}{46}}
                               {{1}{2}{3}{4}{5}}  {{14}{25}{36}}
                                                  {{1}{24}{35}{6}}
                                                  {{13}{2}{46}{5}}
                                                  {{14}{2}{36}{5}}
                                                  {{15}{26}{3}{4}}
                                                  {{1}{25}{3}{4}{6}}
                                                  {{1}{2}{3}{4}{5}{6}}

David Callan, On conjugates for set partitions and integer compositions [math.CO].

Cf. A000110, A000296, A001610, A080107 (self-complementary), A169985, A324012 (self-conjugate), A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
cmp[stn_]:=Union[Sort[Max@@Join@@stn+1-#]&/@stn];
Table[Select[sps[Range[n]],And[cmp[#]==Sort[#],Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]//Length,{n,0,10}]

A324015 Number of nonempty subsets of {1, ..., n} containing no two cyclically successive elements.

Original entry on oeis.org

0, 1, 2, 3, 6, 10, 17, 28, 46, 75, 122, 198, 321, 520, 842, 1363, 2206, 3570, 5777, 9348, 15126, 24475, 39602, 64078, 103681, 167760, 271442, 439203, 710646, 1149850, 1860497, 3010348, 4870846, 7881195, 12752042, 20633238, 33385281, 54018520, 87403802

Offset: 0

Gus Wiseman, Feb 12 2019

nonn

Cyclically successive means 1 succeeds n.

After a(1) = 1, same as A001610 shifted once to the right. Also, a(n) = A169985(n) - 1.

			The a(6) = 17 stable subsets:
  {1}, {2}, {3}, {4}, {5}, {6},
  {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6},
  {1,3,5}, {2,4,6}.

Cf. A000045, A000126, A000296, A001610, A001644, A169985, A306357, A324014.

stabsubs[g_]:=Select[Rest[Subsets[Union@@g]],Select[g,Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&];
Table[Length[stabsubs[Partition[Range[n],2,1,1]]],{n,0,10}]

For n <= 3, a(n) = n. Otherwise, a(n) = a(n - 1) + a(n - 2) + 1.

A330605 a(n) = exp(-1) * Sum_{k>=0} (n*k - 1)^n / k!.

Original entry on oeis.org

1, 0, 5, 89, 2737, 121399, 7316101, 572218716, 56142822849, 6731180810945, 965898950508901, 163116461798211503, 31969444766902475185, 7187057932197297484108, 1834860441330563739401765, 527403671798720265634312349, 169396494914472404237224898305

Offset: 0

Ilya Gutkovskiy, Dec 19 2019

nonn

Cf. A000110, A000296, A124311, A292914, A307066, A330604.

Table[Exp[-1] Sum[(n k - 1)^n/k!, {k, 0, Infinity}], {n, 0, 16}]
Join[{1}, Table[Sum[(-1)^(n - k) Binomial[n, k] n^k BellB[k], {k, 0, n}], {n, 1, 16}]]
Table[n! SeriesCoefficient[Exp[Exp[n x] - x - 1], {x, 0, n}], {n, 0, 16}]

a(n) = n! * [x^n] exp(exp(n*x) - x - 1).

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

A337040 a(n) = exp(-1/4) * Sum_{k>=0} (4k - 1)^n / (4^k k!).

Original entry on oeis.org

1, 0, 4, 16, 112, 896, 8384, 88320, 1032448, 13242368, 184591360, 2773929984, 44641579008, 765196926976, 13905753980928, 266855007453184, 5388980396818432, 114172599765827584, 2530858142594760704, 58556990344729198592, 1411095950792925904896, 35347148031264582270976

Offset: 0

Ilya Gutkovskiy, Aug 12 2020

nonn

Cf. A000296, A003576, A004213, A337038, A337039, A337041, A337042, A337043.

nmax = 21; CoefficientList[Series[Exp[(Exp[4 x] - 1)/4 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 4^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 21}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 4^k BellB[k, 1/4], {k, 0, n}], {n, 0, 21}]

G.f. A(x) satisfies: A(x) = (1 - 4*x + x*A(x/(1 - 4*x))) / (1 - 3*x - 4*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 4*j*x/(1 + x)).
E.g.f.: exp((exp(4*x) - 1) / 4 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 4^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004213(k).
a(n) ~ 4^(n - 1/4) * n^(n - 1/4) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n - 1/4)). - Vaclav Kotesovec, Jun 26 2022

A337041 a(n) = exp(-1/5) * Sum_{k>=0} (5k - 1)^n / (5^k k!).

Original entry on oeis.org

1, 0, 5, 25, 200, 1875, 20625, 256250, 3534375, 53515625, 881468750, 15667578125, 298478828125, 6060493750000, 130542772265625, 2971013486328125, 71193375156250000, 1790666151318359375, 47145509926611328125, 1296156682961425781250, 37129279010879638671875

Offset: 0

Ilya Gutkovskiy, Aug 12 2020

nonn

Cf. A000296, A003577, A005011, A337038, A337039, A337040, A337042, A337043.

nmax = 20; CoefficientList[Series[Exp[(Exp[5 x] - 1)/5 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 5^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 20}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 5^k BellB[k, 1/5], {k, 0, n}], {n, 0, 20}]

G.f. A(x) satisfies: A(x) = (1 - 5*x + x*A(x/(1 - 5*x))) / (1 - 4*x - 5*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 5*j*x/(1 + x)).
E.g.f.: exp((exp(5*x) - 1) / 5 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 5^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A005011(k).
a(n) ~ 5^(n - 1/5) * n^(n - 1/5) * exp(n/LambertW(5*n) - n - 1/5) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^(n - 1/5)). - Vaclav Kotesovec, Jun 26 2022

A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6k - 1)^n / (6^k k!).

Original entry on oeis.org

1, 0, 6, 36, 324, 3456, 43416, 618192, 9778320, 169827840, 3210376032, 65540155968, 1435094563392, 33510354739200, 830486180748672, 21756166766173440, 600339119317643520, 17394883290643709952, 527782830161632077312, 16727350847049194775552

Offset: 0

Ilya Gutkovskiy, Aug 12 2020

nonn

In general, if m >= 1, b <> 0 and e.g.f. = exp(m*exp(b*x) + r*x + s) then a(n) ~ b^n * n^(n + r/b) * exp(n/LambertW(n/m) - n + s) / (m^(r/b) * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n + r/b)). - Vaclav Kotesovec, Jun 28 2022

Cf. A000296, A003578, A005012, A337038, A337039, A337040, A337041, A337043.

nmax = 19; CoefficientList[Series[Exp[(Exp[6 x] - 1)/6 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 6^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 6^k BellB[k, 1/6], {k, 0, n}], {n, 0, 19}]

G.f. A(x) satisfies: A(x) = (1 - 6*x + x*A(x/(1 - 6*x))) / (1 - 5*x - 6*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 6*j*x/(1 + x)).
E.g.f.: exp((exp(6*x) - 1) / 6 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 6^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A005012(k).
a(n) ~ 6^(n - 1/6) * n^(n - 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n - 1/6)). - Vaclav Kotesovec, Jun 26 2022

cp's OEIS Frontend

A346740 Expansion of e.g.f.: exp(exp(x) - 5*x - 1).

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A102286 Total number of odd blocks in all partitions of n-set.

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A261137 Number of set partitions B'_t(n) of {1,2,...,t} into at most n parts, so that no part contains both 1 and t, or both i and i+1 with 1 <= i < t; triangle B'_t(n), t>=0, 0<=n<=t, read by rows.

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A306357 Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.

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A324014 Number of self-complementary set partitions of {1, ..., n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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Mathematica

A324015 Number of nonempty subsets of {1, ..., n} containing no two cyclically successive elements.

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Mathematica

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A330605 a(n) = exp(-1) * Sum_{k>=0} (n*k - 1)^n / k!.

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A337040 a(n) = exp(-1/4) * Sum_{k>=0} (4*k - 1)^n / (4^k * k!).

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A337040 a(n) = exp(-1/4) * Sum_{k>=0} (4k - 1)^n / (4^k k!).

A337041 a(n) = exp(-1/5) * Sum_{k>=0} (5k - 1)^n / (5^k k!).

A337042 a(n) = exp(-1/6) * Sum_{k>=0} (6k - 1)^n / (6^k k!).