A356902 -id:A356902 - cp's OEIS frontend

A356904 E.g.f. satisfies A(x)^2 * log(A(x)) = x * (exp(x) - 1).

1, 0, 2, 3, -32, -175, 2376, 29617, -371440, -9251919, 91421560, 4529155961, -26677647864, -3160004989271, 1541460644192, 2946529440977865, 19556193589426336, -3498019439220155551, -56274505323609293208, 5077223330715030358009

Offset: 0

Seiichi Manyama, Sep 03 2022

sign

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

Cf. A355843, A356797, A356798, A356902.

Cf. A349654.

nmax = 19; A[_] = 1;
Do[A[x_] = Exp[((Exp[x]-1)*x)/A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

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

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

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

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

a(n) = n! * Sum_{k=0..floor(n/2)} (-2*k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-2*k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(2 * x * (exp(x) - 1))/2 ).
E.g.f.: A(x) = ( -2 * x * (1 - exp(x))/LambertW(-2 * x * (1 - exp(x))) )^(1/2).

A357246 E.g.f. satisfies A(x) * log(A(x)) = (1-x) * (exp(x) - 1).

Original entry on oeis.org

1, 1, -2, 5, -49, 497, -6926, 116510, -2325422, 53538315, -1397740279, 40792008435, -1316056239994, 46509292766172, -1786748828967402, 74139054468535061, -3304409577659864305, 157444695280699565069, -7986085592316390890618, 429645521271113815480246

Offset: 0

Seiichi Manyama, Sep 19 2022

sign

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

Cf. A356902, A357243, A357247.

nmax = 19; A[_] = 1;
Do[A[x_] = Exp[-(((Exp[x]-1)*(x-1))/A[x])]+O[x]^(nmax+1)//Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

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

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

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

E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * ((1-x) * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW((1-x) * (exp(x) - 1)) ).
E.g.f.: A(x) = (1-x) * (exp(x) - 1)/LambertW((1-x) * (exp(x) - 1)).

cp's OEIS Frontend

A356904 E.g.f. satisfies A(x)^2 * log(A(x)) = x * (exp(x) - 1).

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A357246 E.g.f. satisfies A(x) * log(A(x)) = (1-x) * (exp(x) - 1).

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