A362771 -id:A362771 - cp's OEIS frontend

A376577 E.g.f. A(x) satisfies A(x) = exp(x(1 + x^2)A(x)).

1, 1, 3, 22, 197, 2256, 32887, 574624, 11690121, 272279296, 7147894571, 208828168704, 6721780173517, 236392313927680, 9019055052313503, 371059593102831616, 16376333449953390353, 771800236449984086016, 38687542995415388643667, 2055303661966287413837824

Offset: 0

Seiichi Manyama, Sep 28 2024

nonn

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

Cf. A362771, A376578.

a(n) = n!*sum(k=0, n\3, (n-2*k+1)^(n-2*k-1)*binomial(n-2*k, k)/(n-2*k)!);

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

a(n) = n! * Sum_{k=0..floor(n/3)} (n-2*k+1)^(n-2*k-1) * binomial(n-2*k,k)/(n-2*k)!.
E.g.f.: exp( -LambertW(-x*(1+x^2)) ).
a(n) ~ sqrt(3 - 2*r*exp(1)) * n^(n-1) / (exp(n-1) * r^n), where r = 0.3314625163779096554724... is the real root of the equation r*(1 + r^2) = exp(-1). - Vaclav Kotesovec, Sep 29 2024

A376578 E.g.f. A(x) satisfies A(x) = exp(x(1 + x^3)A(x)).

Original entry on oeis.org

1, 1, 3, 16, 149, 1656, 22567, 367144, 7020729, 153727408, 3792553451, 104109493104, 3149158146037, 104087591605336, 3732676321011471, 144352230854996056, 5988802115089340273, 265333292818927985376, 12503715930615717645139, 624511093178389790592352

Offset: 0

Seiichi Manyama, Sep 28 2024

nonn

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

Cf. A362771, A376577.

a(n) = n!*sum(k=0, n\4, (n-3*k+1)^(n-3*k-1)*binomial(n-3*k, k)/(n-3*k)!);

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

a(n) = n! * Sum_{k=0..floor(n/4)} (n-3*k+1)^(n-3*k-1) * binomial(n-3*k,k)/(n-3*k)!.
E.g.f.: exp( -LambertW(-x*(1+x^3)) ).
From Vaclav Kotesovec, Oct 10 2024: (Start)
E.g.f.: -LambertW(-x*(1+x^3))/(x*(1+x^3)).
a(n) ~ sqrt(1 + 4*r^3) * n^(n-1) / (exp(n - 3/2) * r^(n - 1/2)), where r = 0.3524487947753533587882657130849651130602232209406... is the positive real root of the equation r*(1 + r^3) = exp(-1). (End)

A378047 E.g.f. satisfies A(x) = (1+x)^2 * exp(x * A(x) / (1+x)).

Original entry on oeis.org

1, 3, 11, 70, 685, 8966, 147769, 2938608, 68509193, 1832813866, 55354862701, 1863179311676, 69163696768093, 2807246931378462, 123685264726805825, 5879149578658117096, 299892675674572370065, 16340561709320173229906, 947234845622653951286485

Offset: 0

Seiichi Manyama, Nov 15 2024

nonn

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

Cf. A362771, A377826.

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

E.g.f.: (1+x)^2 * exp( -LambertW(-x*(1+x)) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(k+2,n-k)/k!.
a(n) ~ (1 + sqrt(1 + 4*exp(-1)))^2 * sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^(n-3) * n^(n-1) / ((sqrt(1 + 4*exp(-1)) - 1)^n * exp(n - 3/2)). - Vaclav Kotesovec, Nov 15 2024

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A376577 E.g.f. A(x) satisfies A(x) = exp(x(1 + x^2)A(x)).

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A376578 E.g.f. A(x) satisfies A(x) = exp(x(1 + x^3)A(x)).

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A378047 E.g.f. satisfies A(x) = (1+x)^2 * exp(x * A(x) / (1+x)).

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A376577 E.g.f. A(x) satisfies A(x) = exp(x*(1 + x^2)*A(x)).

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A376578 E.g.f. A(x) satisfies A(x) = exp(x*(1 + x^3)*A(x)).

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A378047 E.g.f. satisfies A(x) = (1+x)^2 * exp(x * A(x) / (1+x)).

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A376577 E.g.f. A(x) satisfies A(x) = exp(x(1 + x^2)A(x)).

A376578 E.g.f. A(x) satisfies A(x) = exp(x(1 + x^3)A(x)).