A377894 -id:A377894 - cp's OEIS frontend

A377895 E.g.f. satisfies A(x) = (1 + x) * exp(x * A(x)^3).

1, 2, 15, 283, 8057, 313161, 15436735, 922964771, 64910124753, 5250807814753, 480339263735831, 49032749858906067, 5525542086267361801, 681359718334607629409, 91259859216031641999375, 13193464971338727171704611, 2047721360761921797402720545, 339610337568547449759788735553

Offset: 0

Seiichi Manyama, Nov 11 2024

nonn

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

Cf. A377826, A377894.

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

E.g.f.: (1+x) * exp( -LambertW(-3*x*(1+x)^3)/3 ).
E.g.f.: ( -LambertW(-3*x*(1+x)^3)/(3*x) )^(1/3).
a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(3*k+1,n-k)/k!.

A378043 E.g.f. satisfies A(x) = (1+x) * exp( x * (1+x) * A(x)^2 ).

Original entry on oeis.org

1, 2, 13, 190, 4045, 116746, 4251289, 187255174, 9684799961, 575477786674, 38638577549701, 2893159369162030, 239045577899472997, 21604942464613062010, 2120362938300115706513, 224568728344893756230326, 25529660577970226603535793, 3100816199696659908092912866

Offset: 0

Seiichi Manyama, Nov 15 2024

nonn

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

Cf. A378019, A378044.
Cf. A377894, A378045.
Cf. A377894.

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

E.g.f.: (1+x) * exp( -LambertW(-2*x*(1+x)^3)/2 ).

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

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

Original entry on oeis.org

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

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A377895 E.g.f. satisfies A(x) = (1 + x) * exp(x * A(x)^3).

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A378043 E.g.f. satisfies A(x) = (1+x) * exp( x * (1+x) * A(x)^2 ).

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

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