A362776 -id:A362776 - cp's OEIS frontend

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

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)

A362774 E.g.f. satisfies A(x) = exp( x * (1+x)^2 * A(x)^2 ).

Original entry on oeis.org

1, 1, 9, 115, 2265, 59701, 1981513, 79441167, 3736418801, 201790517833, 12309193580841, 837132560820139, 62809405894333321, 5154060532188515325, 459202970647825870313, 44146740571635016905991, 4555272678073789024849377, 502153774773932684443210513

Offset: 0

Seiichi Manyama, May 02 2023

nonn

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

Cf. A361278, A362772.

Cf. A052750, A360547, A362776.

A362774 := proc(n)
    n!*add((2*k+1)^(k-1) * binomial(2*k,n-k)/k!,k=0..n) ;
end proc:
seq(A362774(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

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(2*k,n-k)/k!.

cp's OEIS Frontend

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

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