A378043 -id:A378043 - cp's OEIS frontend

A378045 E.g.f. satisfies A(x) = (1+x) * exp(x * A(x)^2 / (1+x)).

1, 2, 9, 100, 1693, 39046, 1140589, 40379872, 1680490361, 80409242314, 4349556199441, 262478904794140, 17482853419143061, 1274026039224276430, 100830973069183104245, 8612770277501109271576, 789749958006001265241073, 77375794118912255978104978, 8066966112797470401673208089

Offset: 0

Seiichi Manyama, Nov 15 2024

nonn

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

Cf. A377826, A378046.
Cf. A377894, A378043.
Cf. A362773.

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

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

A378044 E.g.f. satisfies A(x) = (1+x) * exp( x * (1+x) * A(x)^3 ).

Original entry on oeis.org

1, 2, 17, 349, 10661, 444161, 23447635, 1500738989, 112954047113, 9777254959729, 956963374613471, 104510139881448797, 12599380858829314093, 1662018439019972570681, 238128379446158082330779, 36825779588890274967294061, 6113887910300601007096973585, 1084611999181162104894547358561

Offset: 0

Seiichi Manyama, Nov 15 2024

nonn

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

Cf. A378019, A378043.

Cf. A377895.

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

E.g.f.: (1+x) * exp( -LambertW(-3*x*(1+x)^4)/3 ).

a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(4*k+1,n-k)/k!.

cp's OEIS Frontend

A378045 E.g.f. satisfies A(x) = (1+x) * exp(x * A(x)^2 / (1+x)).

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