A258922 -id:A258922 - cp's OEIS frontend

A259063 E.g.f.: Series_Reversion( 3x - 2x*exp(x) ).

1, 4, 54, 1208, 37810, 1521252, 74800558, 4346473840, 291409650378, 22142153337500, 1880332153123270, 176486211108436968, 18142303135426278562, 2027140583610836224468, 244622970048028087152990, 31706140285613089502561504, 4392907300768938557656691194, 647905974466168686991684285836

Offset: 1

Paul D. Hanna, Jun 17 2015

nonn

			E.g.f.: A(x) = x + 4*x^2/2! + 54*x^3/3! + 1208*x^4/4! + 37810*x^5/5! + ...
where A(3*x - 2*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 4*x^2 + 54*x^3 + 1208*x^4 + 37810*x^5 + 1521252*x^6 + ...
where F(x)/x = 1/3 + 2/(3-x)^2 + 2^2/(3-2*x)^3 + 2^3/(3-3*x)^4 + 2^4/(3-4*x)^5 +...

G. C. Greubel, Table of n, a(n) for n = 1..329

Cf. A053492, A259064, A259065, A259066, A259062, A258922.

Rest[CoefficientList[InverseSeries[Series[3*x - 2*x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)

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

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A = x + sum(m=1, n, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

O.g.f.: x * Sum_{n>=0}  2^n / (3 - n*x)^(n+1).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) ~ (c/(3*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(3*exp(1)/2). - Vaclav Kotesovec, Jun 19 2015

A370941 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 + 2log(1-x)) ).

Original entry on oeis.org

1, 2, 18, 304, 7668, 259048, 11001968, 563728464, 33857839360, 2333472749376, 181558569560448, 15743501573763456, 1505641080366272640, 157445985444107880960, 17872580693502293022720, 2188829492626563123881472, 287673783237906407512565760

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A258922, A370940.

Cf. A370938.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+2*log(1-x)))/x))

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

a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^k * (n+k)! * |Stirling1(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

A370940 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - 2log(1+x)) ).

Original entry on oeis.org

1, 2, 14, 184, 3612, 94968, 3139088, 125181936, 5851551680, 313874206656, 19006905318528, 1282738818650496, 95477483835672960, 7770589670409684480, 686519279618695022592, 65436589709543394150912, 6693486627002144059422720, 731378220534326743907266560

Offset: 0

Seiichi Manyama, Mar 06 2024

nonn

Index entries for reversions of series

Cf. A258922, A370941.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-2*log(1+x)))/x))

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

a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^k * (n+k)! * Stirling1(n,k).

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

A371296 E.g.f. satisfies A(x) = 1/(3 - 2exp(xA(x)^2)).

Original entry on oeis.org

1, 2, 26, 674, 26682, 1429682, 96867178, 7946279490, 765861255002, 84837503946962, 10621798904563530, 1483378875680954210, 228626616449674796602, 38549099486166110798322, 7058696888173770772536362, 1394913467379909728350803074, 295904373562519633314958421274

Offset: 0

Seiichi Manyama, Mar 18 2024

nonn

Cf. A004123, A258922.

Cf. A367134.

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

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

cp's OEIS Frontend

A259063 E.g.f.: Series_Reversion( 3*x - 2*x*exp(x) ).

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A370941 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + 2*log(1-x)) ).

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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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A370940 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - 2*log(1+x)) ).

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A371296 E.g.f. satisfies A(x) = 1/(3 - 2*exp(x*A(x)^2)).

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A259063 E.g.f.: Series_Reversion( 3x - 2x*exp(x) ).

A370941 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 + 2log(1-x)) ).

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 ).

A370940 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - 2log(1+x)) ).

A371296 E.g.f. satisfies A(x) = 1/(3 - 2exp(xA(x)^2)).