A052894 -id:A052894 - cp's OEIS frontend

A367163 E.g.f. satisfies A(x) = 1 + A(x)^3 * (exp(x*A(x)) - 1).

1, 1, 9, 160, 4367, 161796, 7592593, 431826760, 28875060411, 2220199609420, 193010401410437, 18720726373805952, 2004328775014537111, 234797380878372574276, 29873926565253226992921, 4102473564838214815027576, 604804589755948599369229811

Offset: 0

Seiichi Manyama, Nov 07 2023

nonn

Cf. A000272, A052894, A367162.

a(n) = sum(k=0, n, (n+3*k)!/(n+2*k+1)!*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (n+3*k)!/(n+2*k+1)! * Stirling2(n,k).

A258923 E.g.f. satisfies A(x) = 1/(4 - 3exp(xA(x))).

Original entry on oeis.org

1, 3, 39, 948, 34401, 1671708, 102120555, 7525926516, 650110587933, 64441071121980, 7211152872419151, 899320094627287908, 123696462771198530265, 18603242077944140548428, 3037136136248214142833747, 534943432937469380612083284, 101114708570035662524213928981, 20416341060201627868414787791068

Offset: 0

Paul D. Hanna, Jun 18 2015

nonn

			E.g.f.: A(x) = 1 + 3*x + 39*x^2/2! + 948*x^3/3! + 34401*x^4/4! + 1671708*x^5/5! +...
where A(4*x - 3*x*exp(x)) = 1/(4 - 3*exp(x)).

Cf. A259064, A052894, A258922.

CoefficientList[1/x*InverseSeries[Series[4*x - 3*x*E^x, {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 19 2015 *)

{a(n) = local(A=1); A = (1/x)*serreverse(4*x - 3*x*exp(x +x^2*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A = 1 + (1/x)*sum(m=1, n+1, Dx(m-1, 3^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1+x+x*O(x^n)); A = exp(sum(m=1, n+1, Dx(m-1, 3^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) A(x) = (1/x) * Series_Reversion( 4*x - 3*x*exp(x) ).
(2) A(x) = 1 + (1/x) * Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * (exp(x)-1)^n * x^n / n!.
(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) = A259064(n+1) / (n+1). - Vaclav Kotesovec, Jun 19 2015
a(n) ~ (c/4)^(n+1) * n^(n-1) / (sqrt(c+1) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(4*exp(1)/3). - Vaclav Kotesovec, Jun 19 2015
a(n) = (1/(n+1)!) * Sum_{k=0..n} 3^k * (n+k)! * Stirling2(n,k). - Seiichi Manyama, Mar 06 2024

A370894 Expansion of e.g.f. (1/x) * Series_Reversion( x(3 - exp(2x))/2 ).

Original entry on oeis.org

1, 1, 6, 64, 1016, 21576, 575680, 18525088, 698625408, 30229271680, 1476535180544, 80371762466304, 4824793854177280, 316685993746640896, 22563822118152880128, 1734427247284290015232, 143072322233503079038976, 12606854482934004152303616

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Cf. A052894, A370934.

Cf. A258922.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(3-exp(2*x))/2)/x))

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

a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^(n-k) * (n+k)! * Stirling2(n,k).

a(n) ~ 2^(2*n+1) * LambertW(3*exp(1))^(n+1) * n^(n-1) / (sqrt(1 + LambertW(3*exp(1))) * 3^(n+1) * exp(n) * (LambertW(3*exp(1)) - 1)^(2*n+1)). - Vaclav Kotesovec, Mar 06 2024

A370934 Expansion of e.g.f. (1/x) * Series_Reversion( x(4 - exp(3x))/3 ).

Original entry on oeis.org

1, 1, 7, 84, 1497, 35676, 1067931, 38548980, 1630600677, 79132611420, 4334891782095, 264625534657188, 17815224081030129, 1311349332273617196, 104778837463344022179, 9031998822763725245268, 835500403485829779202557, 82557790782397502710806396

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A052894, A370894.

Cf. A258923.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(4-exp(3*x))/3)/x))

a(n) = sum(k=0, n, 3^(n-k)*(n+k)!*stirling(n, k, 2))/(n+1)!;

a(n) = (1/(n+1)!) * Sum_{k=0..n} 3^(n-k) * (n+k)! * Stirling2(n,k).

A370990 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - x^3(exp(x) - 1)) ).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 120, 210, 201936, 1996344, 12701520, 64865790, 17053788840, 374788816116, 4944496679304, 50034166184730, 6390396135006240, 239770550508132720, 5363062998193560096, 89908444484550625014, 7402557588108228698040

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A052894, A370988, A370989.

Cf. A358014.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^3*(exp(x)-1)))/x))

a(n) = sum(k=0, n\4, (n+k)!*stirling(n-3*k, k, 2)/(n-3*k)!)/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (n+k)! * Stirling2(n-3*k,k)/(n-3*k)!.

A375897 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/2)))^2.

Original entry on oeis.org

1, 2, 12, 122, 1780, 34082, 810740, 23093562, 767175972, 29140904402, 1246366394548, 59292772664666, 3106206974812292, 177715679350850370, 11026719500616041076, 737552919428497318394, 52907911316906095281508, 4051998061642112552244722

Offset: 0

Seiichi Manyama, Sep 01 2024

Cf. A052894, A375898.

Cf. A005649.

Table[2/(n+2)! * Sum[(n + k + 1)!*StirlingS2[n, k], {k, 0, n} ], {n, 0, 20}] (* Vaclav Kotesovec, Aug 27 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(2-exp(x)))/x)^2))

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052894.
E.g.f.: A(x) = ( (1/x) * Series_Reversion(x * (2 - exp(x))) )^2.
a(n) = (2/(n+2)!) * Sum_{k=0..n} (n+k+1)! * Stirling2(n,k).
a(n) ~ LambertW(2*exp(1))^(n+2) * n^(n-1) / (2^(n+1) * exp(n) * sqrt(LambertW(2*exp(1)) + 1) * (LambertW(2*exp(1)) - 1)^(2*n+2)). - Vaclav Kotesovec, Aug 27 2025

A375898 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/3)))^3.

Original entry on oeis.org

1, 3, 21, 234, 3627, 72498, 1780953, 52013118, 1762754655, 68060512458, 2950869169125, 142006584810918, 7513205987292243, 433548334132153698, 27102592662130603857, 1824854382978573444174, 131676307468686605671623, 10137713081262046098901050

Offset: 0

Seiichi Manyama, Sep 01 2024

nonn

Cf. A052894, A375897.

Cf. A226515.

my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(2-exp(x)))/x)^3))

a(n) = 3*sum(k=0, n, (n+k+2)!*stirling(n, k, 2))/(n+3)!;

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052894.
E.g.f.: A(x) = ( (1/x) * Series_Reversion(x * (2 - exp(x))) )^3.
a(n) = (3/(n+3)!) * Sum_{k=0..n} (n+k+2)! * Stirling2(n,k).

A377424 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^4)).

Original entry on oeis.org

1, 1, 11, 253, 9019, 438021, 26992707, 2018069341, 177498369419, 17959376607061, 2055112480694323, 262437681414074541, 36999068388057870651, 5708040382071000644581, 956533539112835413864739, 173022072326584494697760893, 33600521994423195247370822251, 6972639514725247888782370422261

Offset: 0

Seiichi Manyama, Oct 28 2024

nonn

Cf. A000670, A052894, A367134, A367135.

Cf. A377425, A377428.

a(n) = sum(k=0, n, (4*n+k)!*stirling(n, k, 2))/(4*n+1)!;

a(n) = (1/(4*n+1)!) * Sum_{k=0..n} (4*n+k)! * Stirling2(n,k).

A377428 Expansion of e.g.f. (1/x) * Series_Reversion( x*(2 - exp(x))^4 ).

Original entry on oeis.org

1, 4, 56, 1432, 54184, 2734104, 173032680, 13192623448, 1177932112040, 120610734752920, 13935516914366824, 1793837540679492312, 254604546529825454376, 39504947952102355425304, 6652925600854130108675048, 1208610940763303680263653464, 235601431979292206398224418216

Offset: 0

Seiichi Manyama, Oct 28 2024

nonn

Index entries for reversions of series

Cf. A052894, A376389, A376390.

Cf. A377424, A377425.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(2-exp(x))^4)/x))

a(n) = 4*sum(k=0, n, (4*n+k+3)!*stirling(n, k, 2))/(4*n+4)!;

E.g.f. A(x) satisfies A(x) = 1/(2 - exp(x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377424.
a(n) = (4/(4*n+4)!) * Sum_{k=0..n} (4*n+k+3)! * Stirling2(n,k).

A000763 Number of interval orders constructed from n intervals of generic lengths.

Original entry on oeis.org

1, 3, 19, 195, 2831, 53703, 1264467, 35661979, 1173865927, 44218244943, 1877050837355, 88693432799667, 4618194424504623, 262771389992099719, 16223185411792992403, 1080238361814167993739, 77171781603974127429527

Offset: 1

Richard Stanley

Vincenzo Librandi, Jean-François Alcover and Bruno Berselli, Table of n, a(n) for n = 1..100 (up to n = 21 from _Vincenzo Librandi_, up to n = 40 from _Jean-François Alcover_)

Cf. A052894.

seq(n! * coeff(series(exp(int(RootOf(2*Z-_Z*exp(x*_Z)-1)^2, x)), x, n+1), x, n), n = 1..20); # _Vaclav Kotesovec, Mar 21 2016

A000763[max_] := ( e[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 1; c[1] = 1; y[x_] := Sum[d[k]*x^k, {k, 0, max}]; d[0] = 1; d[1] = 1; cc = CoefficientList[ Series[ e'[x]/e[x] - y[x]^2, {x, 0, max}], x]; dd = CoefficientList[ Series[ y[x]*(2 - Exp[x*y[x]]) - 1, {x, 0, max}], x]; eqdd = Thread[dd == 0]; soldd = Solve[ Thread[dd == 0] ]; eqcc = Thread[(cc /. soldd[[1]]) == 0]; solcc = Solve[ Most[eqcc] ] ; solcc /. Rule -> Set; soldd /. Rule -> Set; Table[c[k], {k, 1, max}] *Range[max]! ); Do[A000763[max], {max, 5, 40, 5}]; A000763[40] (* Jean-François Alcover, Jul 23 2013 *)

seq(n)={my(p=serreverse(2*x - x*exp(x + O(x^n)))/x); Vec(serlaplace(exp( intformal(p^2) )))} \\ Andrew Howroyd, Jun 05 2021

E.g.f. E(x) satisfies E'/E = y^2, where y=1+x+5*x^2/2+... is defined by y*(2-exp(x*y))=1.
E.g.f.: exp(int(RootOf(2*_Z-_Z*exp(x*_Z)-1)^2, x)) [in Maple notation].
a(n) ~ c * n^(n-2) / (r^n * exp(n)), where r = 2*(LambertW(2*exp(1))-1)^2 / LambertW(2*exp(1)) = 0.204378273928311464700648197201... and c = 1/((1 - 1/LambertW(2*exp(1))) * exp(1/2)*sqrt(2*(1 + 1/LambertW(2*exp(1))))) = 1.196923669815370203369255598062684... . - Vaclav Kotesovec, Mar 22 2016

More terms from Vladeta Jovovic, Nov 04 2001

cp's OEIS Frontend

A367163 E.g.f. satisfies A(x) = 1 + A(x)^3 * (exp(x*A(x)) - 1).

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A258923 E.g.f. satisfies A(x) = 1/(4 - 3*exp(x*A(x))).

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A370894 Expansion of e.g.f. (1/x) * Series_Reversion( x*(3 - exp(2*x))/2 ).

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A370934 Expansion of e.g.f. (1/x) * Series_Reversion( x*(4 - exp(3*x))/3 ).

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A370990 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^3*(exp(x) - 1)) ).

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A375897 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/2)))^2.

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A375898 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/3)))^3.

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A377424 E.g.f. satisfies A(x) = 1/(2 - exp(x*A(x)^4)).

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A377428 Expansion of e.g.f. (1/x) * Series_Reversion( x*(2 - exp(x))^4 ).

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A258923 E.g.f. satisfies A(x) = 1/(4 - 3exp(xA(x))).

A370894 Expansion of e.g.f. (1/x) * Series_Reversion( x(3 - exp(2x))/2 ).

A370934 Expansion of e.g.f. (1/x) * Series_Reversion( x(4 - exp(3x))/3 ).

A370990 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - x^3(exp(x) - 1)) ).