A385343 -id:A385343 - cp's OEIS frontend

A006228 Expansion of e.g.f. exp(arcsin(x)).

1, 1, 1, 2, 5, 20, 85, 520, 3145, 26000, 204425, 2132000, 20646925, 260104000, 2993804125, 44217680000, 589779412625, 9993195680000, 151573309044625, 2898026747200000, 49261325439503125, 1049085682486400000, 19753791501240753125, 463695871658988800000

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

nonn

L. B. W. Jolley, Summation of Series. 2nd ed., Dover, NY, 1961, p. 150.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Bisections are expansions of sin(arcsinh(x)) and cos(arcsinh(x)).
Bisections are A101927 and A101928.
Row sums of A385343.
Cf. A002019.
Cf. A166741, A166748. - Jaume Oliver Lafont, Oct 24 2009

a:= n-> n!*coeff(series(exp(arcsin(x)), x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 17 2018

Distribute[ CoefficientList[ Series[ E^ArcSin[x], {x, 0, 21}], x] * Table[ n!, {n, 0, 21}]] (* Robert G. Wilson v, Feb 10 2004 *)
With[{nn=30},CoefficientList[Series[Exp[ArcSin[x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Feb 26 2013 *)
Table[FullSimplify[2^(n-2) * (Exp[Pi/2]-(-1)^n*Exp[-Pi/2]) * Gamma[(n-I)/2] * Gamma[(n+I)/2] / Pi], {n, 0, 20}] (* Vaclav Kotesovec, Nov 06 2014 *)

a(n):=(n-1)!*sum((if n=m then 1 else if oddp(n-m) then 0 else sum((-1)^k*(sum(binomial(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!,i,0,floor(j/2))*(-1)^(k-j),j,1,k))*binomial(k+n-1,n-1),k,1,n-m))/(m-1)!,m,1,n); /* Vladimir Kruchinin, Sep 12 2010 */

i even: a_i = Product_{j=1..i/2-1} 1 + 4j^2, i odd: a_i = Product_{j=1..(i-1)/2} 2 + 4j(j-1). - Cris Moore (moore(AT)santafe.edu), Jan 31 2001
a(0)=1, a(1)=1, a(n) = (1+(n-2)^2)*a(n-2) for n >= 2. Jaume Oliver Lafont, Oct 24 2009
a(n) = (n-1)!*sum((if n=m then 1 else if oddp(n-m) then 0 else sum((-1)^k*(sum(C(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i)*C(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!, i=0..floor(j/2))*(-1)^(k-j), j=1..k))*C(k+n-1,n-1), k=1..n-m))/(m-1)!, m=1..n), n>0. - Vladimir Kruchinin, Sep 12 2010
E.g.f.: exp(arcsin(x))=1+2z/(H(0)-z); H(k)=4k+2+z^2*(4k^2+8k+5)/H(k+1), where z=x/((1-x^2)^1/2); (continued fraction). - Sergei N. Gladkovskii, Nov 20 2011
a(n) ~ (exp(Pi/2)-(-1)^n*exp(-Pi/2)) * n^(n-1) / exp(n). - Vaclav Kotesovec, Oct 23 2013
a(n) = 2^(n-2) * (exp(Pi/2)-(-1)^n*exp(-Pi/2)) * GAMMA((n-I)/2) * GAMMA((n+I)/2) / Pi. - Vaclav Kotesovec, Nov 06 2014

More terms from Christian G. Bower

A296675 Expansion of e.g.f. 1/(1 - arcsinh(x)).

Original entry on oeis.org

1, 1, 2, 5, 16, 69, 368, 2169, 14208, 109929, 970752, 8995821, 88341504, 988161069, 12276025344, 154843019169, 2009594658816, 29484826539345, 476778061430784, 7588488203093205, 121001549512310784, 2205431202369899925, 44538441694414110720, 852615914764223422665

Offset: 0

Ilya Gutkovskiy, Dec 18 2017

sign

a(48) is negative. - Vaclav Kotesovec, Jan 26 2020

			1/(1 - arcsinh(x)) = 1 + x/1! + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 69*x^5/5! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A000111, A001818, A006154, A189780, A385343.

a:=series(1/(1-arcsinh(x)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[1/(1 - ArcSinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[1/(1 - Log[x + Sqrt[1 + x^2]]), {x, 0, nmax}], x] Range[0, nmax]!

x='x+O('x^99); Vec(serlaplace(1/(1-log(x+sqrt(1+x^2))))) \\ Altug Alkan, Dec 18 2017

E.g.f.: 1/(1 - log(x + sqrt(1 + x^2))).
a(n) ~ 8*((4 - Pi^2)*sin(Pi*n/2) - 4*Pi*cos(Pi*n/2)) * n^(n-1) / ((4 + Pi^2)^2 * exp(n)). - Vaclav Kotesovec, Dec 18 2017
a(n) = Sum_{k=0..n} k! * i^(n-k) * A385343(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 27 2025

A189780 Expansion of e.g.f. 1/(1 - arcsin(x)).

Original entry on oeis.org

1, 1, 2, 7, 32, 189, 1328, 11019, 104064, 1111641, 13166592, 172006671, 2448559104, 37814647701, 628513744896, 11201565483219, 212867324706816, 4299987047933745, 91950128086450176, 2076040931023605015, 49332990241672003584, 1231115505653454828525, 32183083119025449861120

Offset: 0

Vladimir Kruchinin, May 02 2011

nonn

Cf. A385346, A385347.

Cf. A385343.

CoefficientList[Series[1/(1-ArcSin[t]), {t, 0, 100}], t] Table[
n!, {n, 0, 100}] (* Emanuele Munarini, Nov 23 2015 *)

a(n):=(n-1)!*sum(m*(1+(-1)^(n-m))/2*sum((sum(binomial(k,j)*2^(1-j)*sum((-1)^((n-m)/2-i-j)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!,i,0,floor(j/2)),j,1,k))*binomial(k+n-1,n-1),k,1,n-m),m,1,n-1)+n!;

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-asin(x)))) \\ Seiichi Manyama, Jun 26 2025

a(n)= (n-1)!*sum(m=1..n-1, m*(1+(-1)^(n-m))/2*sum(k=1..n-m (sum(j=1..k, binomial(k,j)*2^(1-j)*sum(i=0..floor(j/2), (-1)^((n-m)/2-i-j)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!)))*binomial(k+n-1,n-1)))+n!, n>0, a(0)=1.
a(n) ~ cos(1) * n! / (sin(1))^(n+1). - Vaclav Kotesovec, Nov 06 2014
a(n) = Sum_{k=0..n} k! * A385343(n,k). - Seiichi Manyama, Jun 26 2025

More terms from Seiichi Manyama, Jun 26 2025

A385369 Expansion of e.g.f. x + sqrt(x^2 + 1).

Original entry on oeis.org

1, 1, 1, 0, -3, 0, 45, 0, -1575, 0, 99225, 0, -9823275, 0, 1404728325, 0, -273922023375, 0, 69850115960625, 0, -22561587455281875, 0, 9002073394657468125, 0, -4348001449619557104375, 0, 2500100833531245335015625, 0, -1687568062633590601135546875, 0

Offset: 0

Seiichi Manyama, Jun 26 2025

Cf. A006228, A177698, A385343.

my(N=30, x='x+O('x^N)); Vec(serlaplace(x+sqrt(x^2+1)))

E.g.f.: exp(arcsinh(x)).
E.g.f. A(x) satisfies A(x) = 1/A(-x).
a(n) = Sum_{k=0..n} i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) = A177698(n-1) for n > 1.
a(2*n+1) = 0 for n > 0.
a(n) = 2^n * n! * binomial((n+1)/2,n)/(n+1).

A385346 Expansion of e.g.f. 1/(1 - 2 * arcsin(x)).

Original entry on oeis.org

1, 2, 8, 50, 416, 4338, 54272, 792402, 13221888, 248206818, 5177131008, 118784695218, 2973171646464, 80619877999698, 2354230063005696, 73657841729314002, 2458203242895507456, 87165684035402711490, 3272629788196529504256, 129696816160868956695090

Offset: 0

Seiichi Manyama, Jun 26 2025

Cf. A189780, A385347.

Cf. A385343.

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

a(n) = Sum_{k=0..n} 2^k * k! * A385343(n,k).

a(n) ~ sqrt(Pi/2) * cos(1/2) * n^(n + 1/2) / (exp(n) * sin(1/2)^(n+1)). - Vaclav Kotesovec, Jun 27 2025

A385347 Expansion of e.g.f. 1/(1 - 3 * arcsin(x)).

Original entry on oeis.org

1, 3, 18, 165, 2016, 30807, 564912, 12085713, 295498368, 8128142667, 248419104768, 8351633349117, 306299582106624, 12169801665625887, 520721224401217536, 23872081186754865513, 1167357853571179216896, 60652216264444277244435, 3336667444310413833732096

Offset: 0

Seiichi Manyama, Jun 26 2025

Cf. A189780, A385346.

Cf. A385343.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*asin(x))))

a(n) = Sum_{k=0..n} 3^k * k! * A385343(n,k).

a(n) ~ sqrt(2*Pi) * cos(1/3) * n^(n + 1/2) / (3 * exp(n) * sin(1/3)^(n+1)). - Vaclav Kotesovec, Jun 27 2025

A385371 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x))^(1/2).

Original entry on oeis.org

1, 1, 3, 14, 93, 804, 8487, 105720, 1520313, 24790800, 451823403, 9101380320, 200808312405, 4816068148800, 124749498365775, 3470782979053440, 103225781141381745, 3268196553960218880, 109745731806193831635, 3895876984699452280320

Offset: 0

Seiichi Manyama, Jun 27 2025

nonn

Cf. A296675, A385372.

Cf. A001147, A385310, A385343, A385367.

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

E.g.f.: 1/(1 - 2 * log(x + sqrt(x^2 + 1)))^(1/2).
a(n) = Sum_{k=0..n} A001147(k) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(1 + exp(1)) * 2^n * n^n / ((exp(1) - 1)^(n + 1/2) * exp(n/2)). - Vaclav Kotesovec, Jun 27 2025

A385372 Expansion of e.g.f. 1/(1 - 3 * arcsinh(x))^(1/3).

Original entry on oeis.org

1, 1, 4, 27, 264, 3369, 52896, 986187, 21293184, 522491697, 14359993344, 436964488443, 14583637923840, 529683272760537, 20798444046458880, 877927319167721067, 39644175780617748480, 1906959640776766940385, 97344936393086594580480, 5255894631271228490720475

Offset: 0

Seiichi Manyama, Jun 27 2025

nonn

Cf. A296675, A385371.

Cf. A007559, A385311, A385343.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*asinh(x))^(1/3)))

E.g.f.: 1/(1 - 3 * log(x + sqrt(x^2 + 1)))^(1/3).
a(n) = Sum_{k=0..n} A007559(k) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (exp(2/3) + 1)^(1/3) * 2^(n + 1/2) * n^(n - 1/6) / (3^(1/3) * Gamma(1/3) * exp(2*n/3) * (exp(2/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Jun 27 2025

A385424 Expansion of e.g.f. exp( -LambertW(-arcsin(x)) ).

Original entry on oeis.org

1, 1, 3, 17, 137, 1465, 19499, 311873, 5829073, 124796081, 3012319315, 80960234577, 2398138520409, 77630951407529, 2726829925494011, 103300796618253825, 4198494172961579169, 182239547736082960737, 8414068749731088539299, 411754575622058760824593

Offset: 0

Seiichi Manyama, Jun 28 2025

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A385426, A385427.

Cf. A277502, A381142, A385343, A385425.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-asin(x)))))

E.g.f. A(x) satisfies A(x) = exp( arcsin(x) * A(x) ).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * A385343(n,k).
a(n) ~ n^(n-1) / (sqrt(cos(exp(-1))) * sin(exp(-1))^(n - 1/2) * exp(n - 3/2)). - Vaclav Kotesovec, Jun 28 2025

A385425 Expansion of e.g.f. exp( -LambertW(-arcsinh(x)) ).

Original entry on oeis.org

1, 1, 3, 15, 113, 1145, 14499, 220703, 3932865, 80342577, 1851286755, 47510525007, 1344106404849, 41562628517865, 1394711974335939, 50480840239135455, 1960392617938419969, 81309789407316485217, 3587373056789171999811, 167762667997938465311247

Offset: 0

Seiichi Manyama, Jun 28 2025

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A001147, A385428.

Cf. A219503, A385343, A385369, A385424.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-asinh(x)))))

E.g.f. A(x) satisfies A(x) = exp( arcsinh(x) * A(x) ).
E.g.f. A(x) satisfies A(x) = ( x + sqrt(x^2 + 1) )^A(x).
a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
From Vaclav Kotesovec, Jun 28 2025: (Start)
a(n) ~ 2^n * exp((exp(-1) - 1)*n + 3/2) * n^(n-1) / (sqrt(1 + exp(2*exp(-1))) * (exp(2*exp(-1)) - 1)^(n - 1/2)).
Equivalently, a(n) ~ n^(n-1) / (sqrt(cosh(exp(-1))) * sinh(exp(-1))^(n - 1/2) * exp(n - 3/2)). (End)

cp's OEIS Frontend

A006228 Expansion of e.g.f. exp(arcsin(x)).

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A296675 Expansion of e.g.f. 1/(1 - arcsinh(x)).

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A189780 Expansion of e.g.f. 1/(1 - arcsin(x)).

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A385369 Expansion of e.g.f. x + sqrt(x^2 + 1).

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A385346 Expansion of e.g.f. 1/(1 - 2 * arcsin(x)).

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A385347 Expansion of e.g.f. 1/(1 - 3 * arcsin(x)).

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A385371 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x))^(1/2).

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A385372 Expansion of e.g.f. 1/(1 - 3 * arcsinh(x))^(1/3).

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A385424 Expansion of e.g.f. exp( -LambertW(-arcsin(x)) ).

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