A355779 -id:A355779 - cp's OEIS frontend

A355842 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)).

1, 0, 2, 3, 44, 210, 3054, 27300, 449952, 6020784, 115381080, 2053568880, 45733246536, 1010390340960, 25916586868704, 680621684914080, 19881379012231680, 603034125051738240, 19833651290982164544, 680927283288289169280, 24953207662252739030400

Offset: 0

Seiichi Manyama, Jul 18 2022

nonn

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

Cf. A052813, A184949, A355779.

nmax = 20; CoefficientList[Series[LambertW[x * Log[1-x]] / (x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 21 2022 *)

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

a(n) = n!*sum(k=0, n\2, (k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, Aug 28 2022

E.g.f.: exp( -LambertW(x * log(1-x)) ).
E.g.f.: LambertW(x * log(1-x)) / (x * log(1-x)).
a(n) ~ sqrt(1 + exp(1)*r^2/(1-r)) * n^(n-1) / (exp(n-1) * r^n), where r = 0.5123112855238643734867005914814802444318611742265... is the positive root of the equation r*log(1-r) = -exp(-1). - Vaclav Kotesovec, Jul 21 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, Aug 28 2022

A356908 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^2.

Original entry on oeis.org

1, 2, -2, 24, -272, 4840, -107496, 2934400, -94501760, 3511914624, -147882696960, 6959217277056, -361941871760256, 20616017227643136, -1276341094954066176, 85337532623368181760, -6128269375791673718784, 470426144496265208979456

Offset: 0

Seiichi Manyama, Sep 03 2022

sign

Robert Israel, Table of n, a(n) for n = 0..350
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349561, A355779, A356909.

S:= series(exp( LambertW(-2 * log(1-x))),x,51):
seq(n!*coeff(S,x,n),n=0..50); # Robert Israel, Sep 18 2022

nmax = 20; A[_] = 1;
Do[A[x_] = (1/(1-x)^2)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

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

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

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

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

a(n) = Sum_{k=0..n} 2^k * (-k+1)^(k-1) * |Stirling1(n,k)|.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-2 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-2 * log(1-x)) ).
E.g.f.: A(x) = -2 * log(1-x)/LambertW(-2 * log(1-x)).

A355786 E.g.f. satisfies A(x) = 1/(1 - 2*x)^(A(x)/2).

Original entry on oeis.org

1, 1, 5, 42, 497, 7620, 143979, 3241406, 84847489, 2534788296, 85170416115, 3180919433802, 130771002469953, 5869920100483452, 285705285804636411, 14989889385040915830, 843420165009747027969, 50664760467069168337680, 3236433107379299238343779

Offset: 0

Seiichi Manyama, Jul 17 2022

nonn

Cf. A052813, A355779.

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

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

E.g.f.: exp( -LambertW(log(1-2*x)/2) ).
a(n) = Sum_{k=0..n} 2^(n-k) * (k+1)^(k-1) * |Stirling1(n,k)|.
From Vaclav Kotesovec, Jul 18 2022: (Start)
E.g.f.: 2*LambertW(log(1-2*x)/2) / log(1-2*x).
a(n) ~ 2^(n - 1/2) * n^(n-1) * exp(3/2 - n + 2*n*exp(-1)) / (exp(2*exp(-1)) - 1)^(n - 1/2). (End)

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A355842 E.g.f. satisfies A(x) = 1/(1 - x)^(x * A(x)).

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A356908 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^2.

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A355786 E.g.f. satisfies A(x) = 1/(1 - 2*x)^(A(x)/2).

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