A370463 -id:A370463 - cp's OEIS frontend

A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

1, 1, 8, 64, 824, 12968, 252720, 5789712, 153169440, 4589004192, 153643615872, 5684390364288, 230307823878144, 10141452865049088, 482259966649655808, 24630247225278881280, 1344614199041549399040, 78137673004382654223360

Offset: 1

Vladeta Jovovic, Oct 18 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..372
Index entries for reversions of series

Cf. A000108, A052851, A052803, A052886, A052895, A006531, A048287, A293604.

Cf. A370462, A370463.

Rest[CoefficientList[Series[(1-Sqrt[1-4*Log[1+x]])/2, {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, May 03 2015 *)

x='x+O('x^50); Vec(serlaplace((1-sqrt(1-4*log(1+x)))/2)) \\ G. C. Greubel, May 24 2017

a(n) = Sum_{k=1..n} Stirling1(n, k)*k!*Catalan(k-1).
a(n) ~ n! / (2*exp(1/8)*sqrt(Pi) * (exp(1/4)-1)^(n-1/2) * n^(3/2)). - Vaclav Kotesovec, May 03 2015
From Seiichi Manyama, Sep 09 2024: (Start)
E.g.f. satisfies A(x) = (log(1 + x)) / (1 - A(x)).
E.g.f.: Series_Reversion( exp(x * (1 - x)) - 1 ). (End)

A370462 E.g.f. satisfies A(x) = log(1 + x)/(1 - A(x))^2.

Original entry on oeis.org

0, 1, 3, 32, 506, 11254, 319486, 11063352, 452075928, 21295486272, 1136180493504, 67720154888352, 4459760039965248, 321592207168637664, 25201588848786782688, 2132592146864957906688, 193806614782424556184320, 18825630812739265968357120

Offset: 0

Seiichi Manyama, Mar 18 2024

nonn

Index entries for reversions of series

Cf. A087138, A370463.

Table[Sum[(3*k - 2)!/(2*k - 1)!*StirlingS1[n, k], {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 19 2024 *)

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

a(n) = Sum_{k=1..n} (3*k-2)!/(2*k-1)! * Stirling1(n,k).
a(n) ~ n^(n-1) / (sqrt(2) * (exp(4/27) - 1)^(n - 1/2) * exp(n + 2/27)). - Vaclav Kotesovec, Mar 19 2024
E.g.f.: Series_Reversion( exp(x * (1 - x)^2) - 1 ). - Seiichi Manyama, Sep 09 2024

A371315 E.g.f. satisfies A(x) = -log(1 - x)/(1 - A(x))^3.

Original entry on oeis.org

0, 1, 7, 110, 2796, 98754, 4469334, 246741984, 16079405784, 1208082769560, 102810760773096, 9774841791650880, 1026870593449179264, 118121793328191431232, 14766518531481521488704, 1993367920121834019649920, 288988424345833831094150016

Offset: 0

Seiichi Manyama, Mar 18 2024

nonn

Index entries for reversions of series

Cf. A052851, A371314.

Cf. A370463.

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

a(n) = Sum_{k=1..n} (4*k-2)!/(3*k-1)! * |Stirling1(n,k)|.

E.g.f.: Series_Reversion( 1 - exp(-x * (1 - x)^3) ). - Seiichi Manyama, Sep 08 2024

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A087138 Expansion of (1-sqrt(1-4*log(1+x)))/2.

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A370462 E.g.f. satisfies A(x) = log(1 + x)/(1 - A(x))^2.

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A371315 E.g.f. satisfies A(x) = -log(1 - x)/(1 - A(x))^3.

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