A361065 -id:A361065 - cp's OEIS frontend

A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

1, 1, 9, 148, 3673, 123276, 5234599, 269262022, 16279709793, 1131627199816, 88926737901031, 7796168316687906, 754414052156289265, 79872584117422215484, 9184299004593618881655, 1139822558262829096519726, 151857077047173825979147969

Offset: 0

Seiichi Manyama, Mar 01 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..327
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361067, A361068, A361069.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

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

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

a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( -LambertW(-3*x/(1-x))/3 ).
E.g.f.: ( -(1-x)/(3*x) * LambertW(-3*x/(1-x)) )^(1/3).
a(n) ~ (1 + 3*exp(1))^(n + 1/2) * n^(n-1) / (3^(3/2) * exp(n + 1/6)). - Vaclav Kotesovec, Mar 02 2023

A361068 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).

Original entry on oeis.org

1, 1, -1, 13, -127, 2101, -41801, 1030177, -29820127, 995977801, -37660751569, 1590847310581, -74242656468575, 3793664894534269, -210656932372422745, 12630986901470435401, -813335155262348743231, 55977540398642247218449

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..360
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361066, A361067, A361069.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x]^2)] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

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

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

a(n) = n! * Sum_{k=0..n} (-2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(2*x/(1-x))/2 ).
E.g.f.: 1 / sqrt( (1-x)/(2*x) * LambertW(2*x/(1-x)) ).
a(n) ~ (-1)^(n+1) * 2^(-3/2) * exp(-1/2) * (2 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024

A361093 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^2) - 1 ).

Original entry on oeis.org

1, 1, 7, 97, 2049, 58541, 2114143, 92419965, 4746108769, 280105517881, 18683156508471, 1389960074426969, 114119472522112225, 10249863809271551973, 999746622121255094479, 105236583967331849218741, 11891012005206169120252737, 1435560112909007680593616625

Offset: 0

Seiichi Manyama, Mar 01 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..334

Cf. A052873, A361094, A361095, A361096, A361097.

Cf. A212722, A361065.

Table[n! * Sum[(2*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 02 2023 *)

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

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

a(n) ~ n^(n-1) / (2 * 3^(1/4) * (2 - sqrt(3))^n * exp((2 - sqrt(3))*n - (sqrt(3) - 1)/2)). - Vaclav Kotesovec, Mar 02 2023

A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

Original entry on oeis.org

1, 1, -3, 40, -719, 18396, -598157, 23713726, -1108701519, 59735988424, -3644505746549, 248358786667674, -18697767289462967, 1541202721786228060, -138046868771541971373, 13351368704222195975206, -1386710317839048140282783, 153939247458296219191539984

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..338
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361066, A361067, A361068.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x]^3)] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

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

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

a(n) = n! * Sum_{k=0..n} (-3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(3*x/(1-x))/3 ).
E.g.f.: 1 / ( (1-x)/(3*x) * LambertW(3*x/(1-x)) )^(1/3).
a(n) ~ (-1)^(n+1) * 3^(-3/2) * exp(-1/3) * (3 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024

A362773 E.g.f. satisfies A(x) = exp( x * (1+x) * A(x)^2 ).

Original entry on oeis.org

1, 1, 7, 79, 1377, 32161, 947623, 33746511, 1410518273, 67714577857, 3672410420871, 222082390164559, 14817864737168353, 1081393797641087841, 85691459902207874471, 7327398378967991154511, 672511583942513406768897, 65943097191889528063033729

Offset: 0

Seiichi Manyama, May 02 2023

nonn

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

Cf. A047974, A362771.

Cf. A361065.

nmax = 20; CoefficientList[Series[Sqrt[LambertW[-2*x * (1+x)]/(-2*x * (1+x))], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 10 2023 *)

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

E.g.f.: exp( -LambertW(-2*x * (1+x))/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(k,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: sqrt(LambertW(-2*x * (1+x))/(-2*x * (1+x))).
a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * 2^(n-1) * n^(n-1) / ((-1 + sqrt(1 + 2*exp(-1)))^n * exp(n-1)). (End)

A361067 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)) ).

Original entry on oeis.org

1, 1, 1, 4, 9, 76, 175, 3606, 833, 354376, -1605249, 65111410, -718371071, 20105327100, -351241054177, 9362931464446, -214514949732735, 6039303900168976, -165679758877120001, 5093296357218337386, -159900268661169533119, 5405435526807425433220

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..413
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361066, A361068, A361069.

nmax = 21; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x])] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

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

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

a(n) = n! * Sum_{k=0..n} (-k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(x/(1-x)) ).
E.g.f.: 1 / ( (1-x)/x * LambertW(x/(1-x)) ).
a(n) ~ -(-1)^n * exp(-1) * (1 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 02 2023

A362775 E.g.f. satisfies A(x) = exp( x/(1-x)^2 * A(x) ).

Original entry on oeis.org

1, 1, 7, 70, 965, 17216, 379207, 9969772, 305154313, 10668593008, 419714689931, 18358646058644, 884070662867053, 46486344447041032, 2650567497877525423, 162908800485532424236, 10737607698626311094033, 755571950776792829919968

Offset: 0

Seiichi Manyama, May 02 2023

nonn

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

Cf. A082579, A362776.

Cf. A052868, A361065.

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

E.g.f.: exp( -LambertW(-x/(1-x)^2) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+k-1,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: -LambertW(-x/(1-x)^2) * (1-x)^2 / x.
a(n) ~ 2^(n + 1/2) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(3/2) * (1 + 2*exp(-1) - sqrt(1 + 4*exp(-1)))^(n - 1/2) * exp(2*n-1)). (End)

A361143 E.g.f. satisfies A(x) = exp( xA(x)^4/(1 - xA(x)^2) ).

Original entry on oeis.org

1, 1, 11, 241, 8105, 370061, 21403675, 1500521485, 123685912817, 11724012791929, 1256517775425131, 150254377493878505, 19833528195709809817, 2864566162751107839493, 449364739762263286489403, 76084967168410028438252101, 13829896583435315152843525985

Offset: 0

Seiichi Manyama, Mar 02 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..315

Cf. A000262, A361142.

Cf. A212722, A361065, A361093.

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

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

A360939 E.g.f. satisfies A(x) = exp( 2xA(x) / (1-x) ).

Original entry on oeis.org

1, 2, 16, 212, 4016, 99952, 3096448, 115063328, 4993598464, 248071645952, 13888585800704, 865481914527232, 59426130052458496, 4458258196636276736, 362864617248019800064, 31848507841521274769408, 2998685833332127139299328, 301504120063370711801724928

Offset: 0

Seiichi Manyama, Mar 04 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..342
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361212.

Cf. A361065.

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

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

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

a(n) = n! * Sum_{k=0..n} 2^k * (k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp ( -LambertW(-2*x/(1-x)) ).
E.g.f.: -(1-x)/(2*x) * LambertW(-2*x/(1-x)).
a(n) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (sqrt(2) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 10 2023

A362776 E.g.f. satisfies A(x) = exp( x/(1-x)^2 * A(x)^2 ).

Original entry on oeis.org

1, 1, 9, 127, 2601, 70981, 2433673, 100697787, 4886085137, 272168650441, 17121437245161, 1200717094233559, 92892754255837561, 7859587210132504653, 721996671783802854377, 71564871858940414914451, 7613407794191946986893857, 865285095267929315207801233

Offset: 0

Seiichi Manyama, May 02 2023

nonn

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

Cf. A082579, A362775.

Cf. A361065.

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

E.g.f.: exp( -LambertW(-2*x/(1-x)^2)/2 ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n+k-1,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: (1-x) * sqrt(-LambertW(-2*x/(1-x)^2) / (2*x)).
a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * n^(n-1) / (sqrt(2) * (-1 + sqrt(1 + 2*exp(-1)))^(3/2) * (-sqrt(1 + 2*exp(-1)) + 1 + exp(-1))^(n - 1/2) * exp(2*n - 1/2)). (End)

cp's OEIS Frontend

A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A361068 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^2) ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A361093 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^2) - 1 ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A362773 E.g.f. satisfies A(x) = exp( x * (1+x) * A(x)^2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A361067 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)) ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A362775 E.g.f. satisfies A(x) = exp( x/(1-x)^2 * A(x) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A361143 E.g.f. satisfies A(x) = exp( x*A(x)^4/(1 - x*A(x)^2) ).

Views

A361143 E.g.f. satisfies A(x) = exp( xA(x)^4/(1 - xA(x)^2) ).

A360939 E.g.f. satisfies A(x) = exp( 2xA(x) / (1-x) ).