A006531 -id:A006531 - cp's OEIS frontend

A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4xexp(x))).

1, 1, 6, 57, 796, 14785, 344046, 9640225, 316255416, 11896233345, 504918768250, 23874754106401, 1244712973780068, 70940791877082049, 4388291507415513894, 292823509879910802465, 20966854494419642792176, 1603540841320336494905089, 130464295561360336835272050

Offset: 0

Ilya Gutkovskiy, Nov 18 2017

nonn

Inverse binomial transform of A194471.

Seiichi Manyama, Table of n, a(n) for n = 0..356

Cf. A000108, A006531, A052895, A194471, A213644, A295239.

a:=series(2/(1+sqrt(1-4*x*exp(x))),x=0,19): seq(n!*coeff(a,x,n),n=0..18); # Paolo P. Lava, Mar 27 2019

nmax = 18; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 18}]

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ Seiichi Manyama, Aug 15 2023

E.g.f.: 1/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - ...))))), a continued fraction.
a(n) ~ sqrt(2*(1 + LambertW(1/4))) * n^(n-1) / ((LambertW(1/4))^n * exp(n)). - Vaclav Kotesovec, Nov 18 2017
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+1,k)/( (2*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A000108(k)/(n-k)!. - Seiichi Manyama, Aug 15 2023

A052895 E.g.f.: (1/2)/(exp(x) - 1) * (1 - (5 - 4*exp(x))^(1/2)).

Original entry on oeis.org

1, 1, 5, 43, 545, 9211, 195305, 4990483, 149371745, 5128125451, 198696086105, 8578228640323, 408387804764945, 21256203702751291, 1200890923560864905, 73191086773679576563, 4786857909878612350145, 334410103752029126714731

Offset: 0

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

Previous name was: A simple grammar.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 871

Cf. A000108, A006531, A251568.

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

CoefficientList[Series[(1/2)/(E^x-1)*(1-(5-4*E^x)^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)
a[n_] = Sum[k! StirlingS2[n, k] CatalanNumber[k], {k, 0, n}];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Jan 15 2018 *)

E.g.f.: (1/2)/(exp(x) - 1)*(1 - (5 - 4*exp(x))^(1/2)).
a(n) = Sum_{k=0..n} k!*Stirling2(n,k)*Catalan(k). - Vladimir Kruchinin, Sep 15 2010
a(n) ~ sqrt(10)*n^(n-1) / (exp(n)*(log(5/4))^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013
E.g.f.: 1/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + (1 - exp(x))/(1 + ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 18 2017
From Peter Bala, Jan 15 2018: (Start)
E.g.f.: C(exp(x) - 1), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for A000108. Cf. A006531.
Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010. For example, modulo 10 the sequence becomes (1, 1, 5, 3, 5, 1, 5, 3, 5, ...), with an apparent period 1, 5, 3, 5 of length 4 = phi(10) beginning at a(1). (End)
O.g.f.: 1 + Sum_{k>=1} A000108(k)*Product_{r=1..k} r*x/(1 - r*x). - Petros Hadjicostas, Jun 12 2020

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

A048287 Number of semiorders on n labeled nodes whose incomparability graph is connected.

Original entry on oeis.org

1, 1, 7, 61, 751, 11821, 226927, 5142061, 134341711, 3975839341, 131463171247, 4803293266861, 192178106208271, 8356430510670061, 392386967808249967, 19788154572706556461, 1066668756919315412431, 61204224384073232815981

Offset: 1

Richard Stanley

			a(3)=7, the seven semiorders being three disjoint points and the disjoint union of a point and a two-element chain (with six labelings).

Robert Israel, Table of n, a(n) for n = 1..373
Mats Granvik, Power series to calculate Lambert W up to infinity

Cf. A000108, A006531.

Cf. A136595, A136590.

A048287 := n -> add((-1)^(n-k-1)*Stirling2(n,k+1)*(2*k)!/k!, k=0..n-1):
seq(A048287(n), n=1..18); # Peter Luschny, Jan 27 2016

Table[Sum[(-1)^(n - k) StirlingS2[n, k] k!*CatalanNumber[k - 1], {k, n}], {n, 20}] (* Michael De Vlieger, Jan 27 2016 *)
Rest[Range[0, 18]! CoefficientList[Series[1 - 2 (1 - Exp[-x]) /(1 - Sqrt[4 Exp[-x] - 3]), {x, 0, 18}], x]] (* Vincenzo Librandi, Jan 28 2016 *)

{a(n)=local(A136590=matrix(n+1,n+1,r,c,if(r>=c,(r-1)!/(c-1)!*polcoeff(log(1+x+x^2 +x*O(x^n))^(c-1),r-1))));(-1)^(n+1)*(A136590^-1)[n+1,2]} \\ Paul D. Hanna, Jan 10 2008

{a(n) = if( n<0, 0, n! * polcoeff( (1 - sqrt(4*exp(-x + x*O(x^n)) - 3)) / 2, n))}; /* Michael Somos, Nov 26 2017 */

{a(n) = if( n<1, 0, n! * polcoeff( serreverse( -log(1 - x + x^2 + x * O(x^n))), n))}; /* Michael Somos, Nov 26 2017 */

E.g.f.: 1-2*(1-exp(-x))/(1-sqrt(4*exp(-x)-3)).
E.g.f.: (1 - sqrt(4*exp(-x) - 3)) / 2. - Michael Somos, Nov 26 2017
a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n, k)*k!*Catalan(k-1). - Vladeta Jovovic, Oct 18 2003
Equals column 1 (unsigned) of triangle A136595, which is the matrix inverse of the triangle A136590 of trinomial logarithmic coefficients. - Paul D. Hanna, Jan 10 2008
E.g.f A(x)=F(exp(x)-1), F(x)=x*A005043(x). - Vladimir Kruchinin, Sep 07 2010
a(n) = (-1)^(n-1) + Sum_{k=1..n-1} binomial(n,k)*a(k)*a(n-k). - Robert Israel, Mar 01 2016
Given e.g.f. =: A(x), then exp(-x) = A(x)^2 - A(x) + 1 = A'(x)*(1 - 2*A(x)). - Michael Somos, Nov 26 2017
a(n) ~ sqrt(3/8) * n^(n-1) / (log(4/3)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Dec 16 2020

More terms from Vladeta Jovovic, Oct 18 2003

A052803 Expansion of e.g.f. (-1 + sqrt(1 + 4log(1-x)))/(2log(1-x)).

Original entry on oeis.org

1, 1, 5, 44, 566, 9674, 207166, 5343456, 161405016, 5591409720, 218592034584, 9521490534720, 457329182411856, 24014921905589328, 1368772939062117936, 84161443919543331840, 5553011951023694408064, 391360838810043628416384, 29342876851060951124158848

Offset: 0

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

Previous name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..360
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 762

Cf. A006531, A086662, A087152.

spec := [S,{C=Cycle(Z),S=Sequence(B),B=Prod(C,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[-1/(2*Log[1-x]) * (1-(1+4*Log[1-x])^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

E.g.f.: (1/2)/log(-1/(-1+x))*(1-(1-4*log(-1/(-1+x)))^(1/2)).
a(n) ~ 2*sqrt(2) * n^(n-1) / (exp(3*n/4) * (exp(1/4)-1)^(n-1/2)). - Vaclav Kotesovec, Sep 30 2013
a(n) = Sum_{k=0..n} (2k)!/(k+1)! * |Stirling1(n,k)|. - Michael D. Weiner, Dec 23 2014
E.g.f.: 1/(1 + log(1-x)/(1 + log(1-x)/(1 + log(1-x)/(1 + log(1-x)/(1 + ...))))), a continued fraction. - Ilya Gutkovskiy, Nov 19 2017

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

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

Original entry on oeis.org

0, 1, -5, 49, -725, 14401, -360005, 10863889, -384415925, 15612336481, -715930020005, 36592369889329, -2062911091119125, 127170577711282561, -8510569547826528005, 614491222512504748369, -47615614242877583230325, 3941408640018910366196641

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..353
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, triassociative operad "Trias".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383988, A383989, A383991.

Composition of A001003 with exp(x)-1.

nn = 17; f[x_] := (1 + 3*x - Sqrt[1 + 6*x + x^2])/(4*x); Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

Original entry on oeis.org

0, 1, -11, 61, -467, 4381, -49091, 643021, -9615827, 161844541, -3026079971, 62243374381, -1396619164787, 33949401567901, -888725861445251, 24926889744928141, -745755560487363347, 23705772035082494461, -797875590555470224931, 28346366547928396344301

Offset: 0

Michael De Vlieger, May 16 2025

Michael De Vlieger, Table of n, a(n) for n = 0..407
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 "FF6".

Cf. A002050, A006531, A084099, A101851, A114285, A225883, A383985, A383986, A383987, A383988, A383995.

nn = 19; f[x_] := x*(1 - 3*x - x^2 + x^3)/(1 + 3*x + x^2 - x^3);
Range[0, nn]! * CoefficientList[Series[f[-(1 - Exp[x])], {x, 0, nn}], x]

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

Original entry on oeis.org

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]

A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4x)) / (2x) + 1 (given by A000108).

Original entry on oeis.org

0, 1, -3, 19, -191, 2661, -47579, 1040047, -26888511, 802727209, -27178685459, 1029077910411, -43086906080063, 1976633329627789, -98597207392040811, 5313105048925173991, -307587436319162110079, 19038773384213189214417, -1254686724727364725716131

Offset: 0

Michael De Vlieger, May 16 2025

sign

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

Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 32, Table 3, diassociative operad "Dias".

Cf. A003725, A006531, A097388, A111884, A112242, A177885, A318215, A383991, A383992, A383993, A383994, A383995. Composition of exp(x)-1 with -A000108(-x).

cp's OEIS Frontend

A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*exp(x))).

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A052895 E.g.f.: (1/2)/(exp(x) - 1) * (1 - (5 - 4*exp(x))^(1/2)).

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A048287 Number of semiorders on n labeled nodes whose incomparability graph is connected.

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A052803 Expansion of e.g.f. (-1 + sqrt(1 + 4*log(1-x)))/(2*log(1-x)).

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A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3*x - sqrt(1+6*x+x^2)) / (4*x) (A001003).

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A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3*x-x^2+x^3) / (1+3*x+x^2-x^3).

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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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A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4xexp(x))).

A052803 Expansion of e.g.f. (-1 + sqrt(1 + 4log(1-x)))/(2log(1-x)).

A383987 Series expansion of the exponential generating function -tridend(-(1-exp(x))) where tridend(x) = (1 - 3x - sqrt(1+6x+x^2)) / (4*x) (A001003).

A383989 Series expansion of the exponential generating function ff6^!(exp(x)-1) where ff6^!(x) = x * (1-3x-x^2+x^3) / (1+3x+x^2-x^3).

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

A383990 Series expansion of the exponential generating function exp(-dend(-x))-1 where dend(x) = (1 - sqrt(1+4x)) / (2x) + 1 (given by A000108).