A371038 -id:A371038 - cp's OEIS frontend

A371039 E.g.f. satisfies A(x) = exp(x^3*A(x)) / (1-x).

1, 1, 2, 12, 72, 480, 4680, 52920, 645120, 9313920, 153014400, 2720995200, 53428636800, 1154333980800, 26847281260800, 671610658118400, 18064388076134400, 517898679679180800, 15763026427487539200, 508612525689235968000, 17329554246181072896000

Offset: 0

Seiichi Manyama, Mar 09 2024

nonn

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

Cf. A352410, A371038.

my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(-x^3/(1-x))/(-x^3)))

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

E.g.f.: LambertW( -x^3/(1-x) ) / (-x^3).

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * binomial(n-2*k,n-3*k)/k!.

A378094 E.g.f. satisfies A(x) = exp( x^2 * A(x) / (1-x) ) / (1-x).

Original entry on oeis.org

1, 1, 4, 24, 204, 2220, 29640, 469560, 8623440, 180306000, 4231815840, 110217270240, 3155551439040, 98529432281280, 3332752472649600, 121416875166787200, 4740431035737196800, 197475789694088505600, 8743499113411321459200, 410050296758706725721600

Offset: 0

Seiichi Manyama, Nov 16 2024

nonn

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

Cf. A352410, A378095.

Cf. A371038.

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

E.g.f.: exp( -LambertW(-x^2/(1-x)^2) )/(1-x).
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * binomial(n,2*k)/k!.
a(n) ~ sqrt(2) * (1 + exp(1/2))^(n + 3/2) * n^(n-1) / exp(n - 1/4). - Vaclav Kotesovec, Nov 16 2024

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A371039 E.g.f. satisfies A(x) = exp(x^3*A(x)) / (1-x).

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A378094 E.g.f. satisfies A(x) = exp( x^2 * A(x) / (1-x) ) / (1-x).

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