A360474 -id:A360474 - cp's OEIS frontend

A360473 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^2 ).

1, 1, 7, 82, 1441, 34036, 1013149, 36446698, 1538703457, 74607811048, 4086635087701, 249593193648646, 16819085803158577, 1239637405609740268, 99206330021667838285, 8567230421555333516746, 794104205843228382969409

Offset: 0

Seiichi Manyama, Feb 08 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..344
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A273954, A360465, A360474.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x*Exp[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) = sum(k=0, n, k^(n-k)*(2*k+1)^(k-1)*binomial(n, k));

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

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

my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(sum(k=0, N, (k+1)^(k-1)*(2*x*exp(x))^k/k!))))

a(n) = Sum_{k=0..n} k^(n-k) * (2*k+1)^(k-1) * binomial(n,k).
E.g.f.: A(x) = exp( -LambertW(-2*x * exp(x))/2 ).
E.g.f.: A(x) = sqrt( -LambertW(-2*x * exp(x)) / (2*x * exp(x)) ).
E.g.f.: A(x) = sqrt( Sum_{k>=0} (k+1)^(k-1) * (2*x * exp(x))^k / k! ).
a(n) ~ sqrt(1 + LambertW(exp(-1)/2)) * n^(n-1) / (2 * exp(n - 1/2) * LambertW(exp(-1)/2)^n). - Vaclav Kotesovec, Feb 17 2023

A380425 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * exp(x * A(x)^(1/2)) ).

Original entry on oeis.org

1, 2, 12, 116, 1592, 28472, 630028, 16649348, 512197456, 17993496176, 711065689364, 31231930472492, 1509776777566648, 79670350504209896, 4557716010219325468, 280992142281969312548, 18574365176584473753248, 1310583528463442480750048, 98318677221689347734929956

Offset: 0

Seiichi Manyama, Jan 24 2025

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A162695, A380426, A380427.

Cf. A360474, A380406.

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A162695.

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

A360471 E.g.f. satisfies A(x) = x * exp( 2A(x) + x exp(2*A(x)) ).

Original entry on oeis.org

0, 1, 6, 75, 1476, 39805, 1366278, 56998179, 2800588808, 158420939193, 10140538486410, 724652822705119, 57187947315670284, 4939834587311520117, 463572330418586227790, 46965096302630022564315, 5108915146530700018466832, 593925863391217441843199089

Offset: 0

Seiichi Manyama, Feb 09 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..337

Cf. A055779, A360442, A360474, A360481.

a(n) = sum(k=1, n, k^(n-k)*(2*n)^(k-1)*binomial(n, k));

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

a(n) ~ 2^(n - 1/2) * s^n * n^(n-1) / (sqrt(2 + 1/s - 4*s) * (1 - 2*s)^n * exp(n*(1 - 2*s))), where s = 0.3875920123187127910093095185777835252050660050582... is the root of the equation 2*s*(1 + LambertW(s)) = 1. - Vaclav Kotesovec, Feb 17 2023

A380879 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2xexp(x)) ).

Original entry on oeis.org

1, 2, 16, 230, 4888, 138442, 4916140, 210270734, 10530743632, 604747157138, 39185881490644, 2828691317839510, 225137088955561144, 19588316964130880474, 1849745928662841982588, 188421660506420000503838, 20594905554562935801454240, 2404374864844251715105658146

Offset: 0

Seiichi Manyama, Feb 07 2025

nonn

Index entries for reversions of series

Cf. A162695, A380880.

Cf. A360474.

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

E.g.f. A(x) satisfies A(x) = exp(2 * x * A(x) * exp(x * A(x))).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A360474.
a(n) = 2 * Sum_{k=0..n} k^(n-k) * (2*n+2)^(k-1) * binomial(n,k).

A380881 E.g.f. A(x) satisfies A(x) = exp( x * A(x)^3 * exp(x * A(x)^3) ).

Original entry on oeis.org

1, 1, 9, 163, 4541, 171781, 8231395, 478055299, 32642065433, 2562896897353, 227510655792191, 22533214047347455, 2463465770439307045, 294676777871863052173, 38284087227668033391515, 5368383942726216941810971, 808133883137288259018215345, 129988823008132636178027546257

Offset: 0

Seiichi Manyama, Feb 07 2025

nonn

Index entries for reversions of series

Cf. A162695, A360474.

Cf. A362656, A380880.

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

E.g.f.: ( (1/x) * Series_Reversion( x * exp(-3*x*exp(x)) ) )^(1/3).

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

cp's OEIS Frontend

A360473 E.g.f. satisfies A(x) = exp( x * exp(x) * A(x)^2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

PARI

Formula

A380425 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * exp(x * A(x)^(1/2)) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A360471 E.g.f. satisfies A(x) = x * exp( 2*A(x) + x * exp(2*A(x)) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A380879 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x*exp(x)) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A380881 E.g.f. A(x) satisfies A(x) = exp( x * A(x)^3 * exp(x * A(x)^3) ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A360471 E.g.f. satisfies A(x) = x * exp( 2A(x) + x exp(2*A(x)) ).

A380879 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2xexp(x)) ).