A002050 -id:A002050 - cp's OEIS frontend

A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

0, 1, -1, 4, -23, 181, -1812, 22037, -315569, 5201602, -97009833, 2019669961, -46432870222, 1168383075471, -31939474693297, 942565598033196, -29866348653695203, 1011335905644178273, -36446897413531401020, 1392821757824071815641, -56259101478392975833333

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..200
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, permutative commutative operad "Perm/Com2".

Cf. A002050, A006531, A084099, A101851, A114285, A177885, A225883, A383986, A383987, A383988, A383989.

Composition of A000169 with signs and 1-exp(x).

nn = 20; f[x_] := -Sum[k^(k - 1)*(1 - Exp[x])^k/k!, {k, nn}];
Range[0, nn]! * CoefficientList[Series[f[x], {x, 0, nn}], x]

A383986 Expansion of the exponential generating function sqrt(4exp(x) - exp(2x) - 2) - 1.

Original entry on oeis.org

0, 1, -1, 1, -13, 61, -601, 5881, -73333, 1021861, -16334401, 290146561, -5707536253, 122821558861, -2873553719401, 72586328036041, -1969306486088773, 57106504958139061, -1762735601974347601, 57705363524117482321, -1996916624448159410893

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..409
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, operad "NAC_2".

Cf. A002050, A006531, A084099, A101851, A114285, A182037, A225883, A383985, A383987, A383988, A383989.

nn = 20; f[x_] := -1 + Sqrt[1 + 2 x - x^2];
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383988 Series expansion of the exponential generating function -postLie(1-exp(x)) where postLie(x) = -log((1 + sqrt(1-4*x)) / 2) (given by A006963).

Original entry on oeis.org

0, 1, -2, 12, -110, 1380, -22022, 426972, -9747950, 256176660, -7617417302, 252851339532, -9268406209790, 371843710214340, -16206868062692582, 762569209601624892, -38525315595630383630, 2079964082064837282420, -119513562475103977951862

Offset: 0

Michael De Vlieger, May 16 2025

The series -postLie(-x) is the inverse for the substitution of the series comTrias(x), given by the suspension of the Koszul dual of comTrias. - Bérénice Delcroix-Oger, May 28 2025

Michael De Vlieger, Table of n, a(n) for n = 0..374
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28, Table 2, commutative triassociative operad "ComTrias".

Cf. A002050, A006531, A084099, A097388, A101851, A114285, A225883, A383985, A383986, A383987, A383989. Composition of -A006963(-x) and exp(x)-1.

nn = 18; f[x_] := Log[(1 + Sqrt[1 + 4*x])/2];
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A032109 "BIJ" (reversible, indistinct, labeled) transform of 1,1,1,1,...

Original entry on oeis.org

1, 1, 2, 7, 38, 271, 2342, 23647, 272918, 3543631, 51123782, 811316287, 14045783798, 263429174191, 5320671485222, 115141595488927, 2657827340990678, 65185383514567951, 1692767331628422662, 46400793659664205567, 1338843898122192101558, 40562412499252036940911

Offset: 0

Christian G. Bower

nonn

Starting (1, 2, 7, 38, 271, ...) = A008292 * [1, 1, 2, 4, 8, ...]. - Gary W. Adamson, Dec 25 2008
The inverse binomial transform is 1, 0, 1, 3, 19, 135, 1171, 11823, 136459, ..., see A091346. - R. J. Mathar, Oct 17 2012
Stirling transform of A001710. - Anton Zakharov, Aug 06 2016
For n even (resp. n odd), a(n) counts the ordered partitions of {1,2,...,n} with an even (resp. odd) number of blocks, and a(n)-1 counts the ordered partitions of {1,2,...,n} with an odd (resp. even) number of blocks. - Jose A. Rodriguez, Feb 04 2021

Alois P. Heinz, Table of n, a(n) for n = 0..424 (first 101 terms from R. J. Mathar)
C. G. Bower, Transforms (2)

A000629, A000670, A002050, A032109, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012
A052856(n)=2*a(n), if n>0.
Cf. A008292, A001710.

A032109 := proc(n)
    (A000670(n)+1)/2 ;
end proc: # R. J. Mathar, Oct 17 2012
a := n -> (polylog(-n, 1/2)+`if`(n=0,3,2))/4:
seq(round(evalf(a(n), 32)), n=0..18); # Peter Luschny, Nov 03 2015
# alternative Maple program:
b:= proc(n, m) option remember; `if`(n=0, m!,
      add(b(n-1, max(m, j)), j=1..m+1))
    end:
a:= n-> (b(n,0)+1)/2:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

Table[(PolyLog[-n, 1/2] + 2 + KroneckerDelta[n])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 02 2015 *)

a(n)=if(n<0,0,n!*polcoeff(subst((1-y^2/2)/(1-y),y,exp(x+x*O(x^n))-1),n))

list(n)=my(v=Vec(subst((1-y^2/2)/(1-y),y,exp(x+x*O(x^n))-1)));vector(n+1,i,v[i]*(i-1)!) \\ Charles R Greathouse IV, Oct 17 2012

E.g.f.: (e^(2*x)-2*e^x-1)/(2*e^x-4).
a(n) = (A000670(n)+1)/2. - Vladeta Jovovic, Apr 13 2003
a(n) = A052875(n)/2 + 1. - Max Alekseyev, Jan 31 2021
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/(2*log(2)^(n+1)*exp(n)). - Ilya Gutkovskiy, Aug 06 2016
a(n) = Sum_{s in S_n^even} Product_{i=1..n} binomial(i,s(i)-1), where s ranges over the set S_n^even of even permutations of [n]. - Jose A. Rodriguez, Feb 02 2021

A006485 a(n) = (2^(2^n + 1) + 1)/3.

Original entry on oeis.org

3, 11, 171, 43691, 2863311531, 12297829382473034411, 226854911280625642308916404954512140971, 77194726158210796949047323339125271902179989777093709359638389338608753093291

Offset: 1

Dennis S. Kluk (mathemagician(AT)ameritech.net)

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Delbert L. Johnson, Table of n, a(n) for n = 1..11
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979.

Cf. A001045, A070969.

lst={};Do[AppendTo[lst, (2^(2^n+1)+1)/3], {n, 9}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2008 *)
Table[(1+2^(2^n+1))/3,{n,10}] (* Harvey P. Dale, Apr 03 2023 *)

{a(n) = if(n > 0, (2^(2^n + 1) + 1)/3, 0)}; /* Michael Somos, Mar 30 2020 */

a(n) = A001045(2^n+1) = (3*a(n-1)^2 + 1)/2 - a(n-1). - Michael Somos, Mar 30 2020

a(n) = A070969(n)/3. - Alois P. Heinz, Mar 28 2023

A054255 Triangle T(n,k) (n >= 1, 0<=k<=n) giving number of preferential arrangements of n things beginning with k (transposed, then read by rows).

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 6, 18, 25, 26, 24, 84, 134, 149, 150, 120, 480, 870, 1050, 1081, 1082, 720, 3240, 6600, 8700, 9302, 9365, 9366, 5040, 25200, 57120, 82320, 92526, 94458, 94585, 94586, 40320, 221760, 554400, 871920, 1038744, 1085364, 1091414, 1091669, 1091670

Offset: 1

Eugene McDonnell (Eemcd(AT)aol.com), May 05 2000

Can be generated from Stirling_2 triangle A008277 (cf. A028246, which is intermediate between the two arrays).

			   1;
   1,  2;
   2,  5,   6;
   6, 18,  25,  26;
  24, 84, 134, 149, 150;
  ...

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Row sums give A000670. First 3 rows are A000629, A002050 = A000629 - 1, 2*A002051 = (A000629 - 2^m) (m >= 0).

Cf. A090665 (triangle with rows reversed).

More terms from James Sellers, May 05 2000

A000154 Erroneous version of A003713.

Original entry on oeis.org

1, 1, 2, 7, 35, 228, 1834, 17382, 195866, 2487832, 35499576, 562356672, 9794156448, 186025364016, 3826961710272, 84775065603888, 2011929826983504

Offset: 0

dead

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. S. Kluk & N. J. A. Sloane, Correspondence, 1979

A052877 E.g.f.: exp(x)-1+log(-1/(-2+exp(x))).

Original entry on oeis.org

0, 2, 3, 7, 27, 151, 1083, 9367, 94587, 1091671, 14174523, 204495127, 3245265147, 56183135191, 1053716696763, 21282685940887, 460566381955707, 10631309363962711, 260741534058271803, 6771069326513690647

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 848

a(n) = A000629(n-1)+1, n>0. Cf. A002050.

spec := [S,{B=Set(Z,1 <= card),C=Cycle(B),S=Union(B,C)},labeled]: seq(combstruct[count](spec, size=n), n=0..20);

CoefficientList[Series[E^x-1+Log[-1/(-2+E^x)], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

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

a(n) ~ (n-1)! / log(2)^n. - Vaclav Kotesovec, Sep 30 2013

New name, using e.g.f., from Vaclav Kotesovec, Sep 30 2013

cp's OEIS Frontend

A383985 Series expansion of the exponential generating function LambertW(1-exp(x)), see A000169.

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A383986 Expansion of the exponential generating function sqrt(4*exp(x) - exp(2*x) - 2) - 1.

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A383988 Series expansion of the exponential generating function -postLie(1-exp(x)) where postLie(x) = -log((1 + sqrt(1-4*x)) / 2) (given by A006963).

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A032109 "BIJ" (reversible, indistinct, labeled) transform of 1,1,1,1,...

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PARI

PARI

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A006485 a(n) = (2^(2^n + 1) + 1)/3.

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A054255 Triangle T(n,k) (n >= 1, 0<=k<=n) giving number of preferential arrangements of n things beginning with k (transposed, then read by rows).

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A000154 Erroneous version of A003713.

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A052877 E.g.f.: exp(x)-1+log(-1/(-2+exp(x))).

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A383986 Expansion of the exponential generating function sqrt(4exp(x) - exp(2x) - 2) - 1.