A362771 -id:A362771 - cp's OEIS frontend

A376492 E.g.f. satisfies A(x) = exp(x^2 * (1 + x) * A(x)^2).

1, 0, 2, 6, 60, 600, 7680, 123480, 2212560, 47053440, 1104092640, 29200802400, 845985349440, 26864561243520, 924556913280000, 34334318184566400, 1367790957223891200, 58194757879908249600, 2633788044958380710400, 126340003102675832870400

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

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

Cf. A362771, A376493.

Cf. A376476.

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

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

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

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

A376493 E.g.f. satisfies A(x) = exp(x^3 * (1 + x) * A(x)^3).

Original entry on oeis.org

1, 0, 0, 6, 24, 0, 2520, 35280, 141120, 6048000, 181440000, 1995840000, 51831964800, 2280127449600, 47882676441600, 1192991325926400, 59048471978496000, 1942527607308288000, 56983429057076121600, 2842216483159788134400, 126830901998902413312000

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

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

Cf. A362771, A376492.

Cf. A376477.

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

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

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

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

A363356 E.g.f. satisfies A(x) = exp(x * A(x) * (1 + x * A(x)^3)).

Original entry on oeis.org

1, 1, 5, 52, 821, 17536, 473497, 15476224, 594230345, 26221431808, 1307680266221, 72739285725184, 4465197522732157, 299855584017743872, 21867349264346912705, 1721013285639521959936, 145394112130209844644113, 13123788855563296766427136

Offset: 0

Seiichi Manyama, Aug 17 2023

nonn

Cf. A088695, A362771, A363355.

Cf. A365013.

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

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

A372154 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x) * A(x)^(1/2) ).

Original entry on oeis.org

1, 2, 12, 98, 1128, 16442, 293356, 6195114, 151432112, 4209004466, 131188519764, 4533821784098, 172125130420744, 7122734349079338, 319148172778019708, 15395906192167996058, 795673541794111734624, 43862837291529529270370

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

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

Cf. A362771, A372155.

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

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

E.g.f.: A(x) = exp( -2 * LambertW(-x * (1+x)) ).
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(s*k,n-k)/k!.
a(n) ~ sqrt(2 + 8*exp(-1) - 2*sqrt(1 + 4*exp(-1))) * 2^n * n^(n-1) / ((sqrt(1+4*exp(-1)) - 1)^n * exp(n - 5/2)). - Vaclav Kotesovec, Aug 05 2025

A372155 E.g.f. A(x) satisfies A(x) = exp( 3 * x * (1 + x) * A(x)^(1/3) ).

Original entry on oeis.org

1, 3, 21, 198, 2505, 39348, 743967, 16465494, 418281393, 12006610344, 384595471119, 13607063765298, 527217367699881, 22209587195328588, 1010947593782034687, 49457001919808733102, 2588247541696766293857, 144302243002459116148944

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

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

Cf. A362771, A372154.

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

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

E.g.f.: A(x) = exp( -3 * LambertW(-x * (1+x)) ).

If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s ), then a(n) = r * n! * Sum_{k=0..n} (t*k+u*(n-k)+r)^(k-1) * binomial(s*k,n-k)/k!.

A372162 E.g.f. A(x) satisfies A(x) = exp( x * sqrt(1+2x) A(x) ).

Original entry on oeis.org

1, 1, 5, 31, 329, 4201, 70357, 1374703, 31888817, 839198737, 25021698821, 827967913279, 30240609486265, 1205630521463161, 52177446181578005, 2434309587346377871, 121857094322821338593, 6513265883385904609057, 370302655720337288548741

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

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

Cf. A362771, A372163.

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

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

E.g.f.: A(x) = exp( -LambertW(-x * sqrt(1+2*x)) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * 2^(n-k) * binomial(k/2,n-k)/k!.
From Vaclav Kotesovec, Apr 21 2024: (Start)
E.g.f.: -LambertW(-x*sqrt(1 + 2*x))/(x*sqrt(1 + 2*x)).
a(n) ~ sqrt(3*r + 1) * n^(n-1) / ((1 + 2*r)^(3/4) * exp(n - 1/2) * r^(n + 1/2)), where r = (exp(2/3) + (-exp(1) + (6*(9 + sqrt(81 - 3*exp(2))))/exp(1))^(2/3)) / (6*(54 - exp(2) + 6*sqrt(81 - 3*exp(2)))^(1/3)) - 1/6 = 0.292252770550601628... (End)

A372163 E.g.f. A(x) satisfies A(x) = exp( x * (1+3x)^(1/3) A(x) ).

Original entry on oeis.org

1, 1, 5, 28, 321, 3636, 65947, 1154238, 28622001, 684987400, 21513702771, 656705784714, 24936869827465, 928288327257084, 41315505985090443, 1817727059210127286, 92749773791662574433, 4712674616532693996432, 271157923143678988333027

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

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

Cf. A362771, A372162.

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

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

E.g.f.: A(x) = exp( -LambertW(-x * (1+3*x)^(1/3)) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * 3^(n-k) * binomial(k/3,n-k)/k!.
a(n) ~  sqrt(4*r+1) * n^(n-1) / (exp(n - 5/2) * r^(n - 3/2)), where r = 0.29742497866288781360719311656731644994668261137281157848090655000... is the root of the equation r*(1 + 3*r)^(1/3) = exp(-1). - Vaclav Kotesovec, Apr 22 2024

A376146 E.g.f. satisfies A(x) = exp( x * (1+x)^4 * A(x) ).

Original entry on oeis.org

1, 1, 11, 124, 1997, 42616, 1120327, 35203960, 1288741337, 53898829408, 2536932089771, 132770439164584, 7649993702503429, 481295935534882768, 32834728249861856879, 2414570451161244199576, 190412665638185073399473, 16030575396743899522805440

Offset: 0

Seiichi Manyama, Sep 11 2024

nonn

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

Cf. A362771, A362772, A376145.

Cf. A360082, A367790.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x)^4))))

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

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

A376517 E.g.f. satisfies A(x) = exp(x^2 * (1 + x) * A(x)).

Original entry on oeis.org

1, 0, 2, 6, 36, 360, 3000, 40320, 532560, 8527680, 152591040, 2987107200, 65408333760, 1544664401280, 39767121313920, 1100734899264000, 32661264290054400, 1034874195222067200, 34834463447361177600, 1242657968679512985600, 46804841790705090892800

Offset: 0

Seiichi Manyama, Sep 26 2024

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

Cf. A362771, A376518.

Cf. A376492, A376512.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x^2*(1+x)))))

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

E.g.f.: exp( -LambertW(-x^2 * (1+x)) ).
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * binomial(k,n-2*k)/k!.
a(n) ~ sqrt((2 + 3*r)/(1 + r)) * n^(n-1) / (exp(n-1) * r^n), where r = (-1 + 2*cosh(log(-1 + (3*(9 + sqrt(81 - 12*exp(1))))/(2*exp(1)))/3))/3. - Vaclav Kotesovec, Sep 26 2024

A376518 E.g.f. satisfies A(x) = exp(x^3 * (1 + x) * A(x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 0, 1080, 15120, 60480, 967680, 29030400, 319334400, 3772137600, 129729600000, 2724321600000, 41366099174400, 1238803517952000, 38414242234368000, 840907325318860800, 23606245443503923200, 878145842759657472000, 26509751796795531264000

Offset: 0

Seiichi Manyama, Sep 26 2024

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

Cf. A362771, A376517.

Cf. A376493, A376513.

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

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

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

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

cp's OEIS Frontend

A376492 E.g.f. satisfies A(x) = exp(x^2 * (1 + x) * A(x)^2).

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A376493 E.g.f. satisfies A(x) = exp(x^3 * (1 + x) * A(x)^3).

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A363356 E.g.f. satisfies A(x) = exp(x * A(x) * (1 + x * A(x)^3)).

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A372154 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x) * A(x)^(1/2) ).

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A372155 E.g.f. A(x) satisfies A(x) = exp( 3 * x * (1 + x) * A(x)^(1/3) ).

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A372162 E.g.f. A(x) satisfies A(x) = exp( x * sqrt(1+2*x) * A(x) ).

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A372163 E.g.f. A(x) satisfies A(x) = exp( x * (1+3*x)^(1/3) * A(x) ).

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A376146 E.g.f. satisfies A(x) = exp( x * (1+x)^4 * A(x) ).

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A376517 E.g.f. satisfies A(x) = exp(x^2 * (1 + x) * A(x)).

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A372162 E.g.f. A(x) satisfies A(x) = exp( x * sqrt(1+2x) A(x) ).

A372163 E.g.f. A(x) satisfies A(x) = exp( x * (1+3x)^(1/3) A(x) ).