A361182 -id:A361182 - cp's OEIS frontend

A360609 E.g.f. satisfies A(x) = exp(x*A(x)^3) / (1-x).

1, 2, 17, 313, 9053, 357941, 17975605, 1095604133, 78570635225, 6482415935449, 604889610870881, 62989604872166897, 7241672622495518773, 911048848278644776949, 124497704904842673086285, 18364053909500922198147421, 2908158473059042016441887025

Offset: 0

Seiichi Manyama, Mar 05 2023

nonn

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

Cf. A352410, A360601.
Cf. A052752, A361182.
Cf. A370876.

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

E.g.f.: (LambertW( -3*x/(1-x)^3 ) / (-3*x))^(1/3).
a(n) ~ 3^(-5/6) * (2^(4/3) + 2*(3 + sqrt(4*exp(1) + 9))^(1/3) * exp(-2/3) - 2^(2/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3))^(1/6) * 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(4/9) * sqrt(4 - 2^(4/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + 3*2^(2/3) * exp(-2/3) * (3 + sqrt(4*exp(1) + 9))^(1/3)) * n^(n-1) * (12 + 4*sqrt(4*exp(1) + 9))^(n/3) / (exp(7/18 + 5*n/3) * (2 - 2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) + exp(-2/3) * (12 + 4*sqrt(4*exp(1) + 9))^(1/3))^n * ((3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2^(2/3))^(3/2) * sqrt(2^(1/3) * (3 + sqrt(4*exp(1) + 9))^(2/3) * exp(-1/3) - 2)). - Vaclav Kotesovec, Mar 06 2023
a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n+2*k,n-k)/k!. - Seiichi Manyama, Mar 09 2024

A361193 E.g.f. satisfies A(x) = exp( -2xA(x) ) / (1-x).

Original entry on oeis.org

1, -1, 6, -50, 648, -10952, 232336, -5919664, 176435328, -6024464000, 231972167424, -9946181374208, 470038191434752, -24276240445152256, 1360508977539004416, -82233680186863536128, 5332689963474238341120, -369321737420738845638656

Offset: 0

Seiichi Manyama, Mar 03 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..360
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A352410, A352448, A361182, A361194.

Cf. A361068.

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

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

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

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

A361194 E.g.f. satisfies A(x) = exp( -3xA(x) ) / (1-x).

Original entry on oeis.org

1, -2, 17, -237, 4893, -133683, 4567905, -187666587, 9017657433, -496470972951, 30824023641669, -2131090659947439, 162397790115179733, -13525005928296072915, 1222285110682680848169, -119135392516302191619507, 12458374493322416970025521

Offset: 0

Seiichi Manyama, Mar 03 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..338
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A352410, A352448, A361182, A361193.

Cf. A361069.

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

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

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

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

A380826 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3x) / (1 + xexp(-2*x)) ).

Original entry on oeis.org

1, 4, 43, 810, 22273, 811728, 36979467, 2025462736, 129748802401, 9522843081984, 788169731306059, 72641846664240384, 7379343546762675873, 819269203286474309632, 98698960328223628470379, 12824232015954542746048512, 1787731339345567827140060737, 266157254062414638948185210880

Offset: 0

Seiichi Manyama, Feb 04 2025

nonn

Index entries for reversions of series

Cf. A088690, A161633, A380808.

Cf. A361182, A380829, A380830.

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

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

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

A380830 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x) ).

Original entry on oeis.org

1, 4, 47, 978, 29769, 1201728, 60656679, 3681441648, 261337079601, 21256149703680, 1949700750690879, 199146039242552064, 22420399033075845177, 2758645779752490872832, 368321963942753147683575, 53038788218443786432223232, 8194316429830951008255159009, 1352065789150879084276947222528

Offset: 0

Seiichi Manyama, Feb 05 2025

nonn

Index entries for reversions of series

Cf. A361182, A380826, A380829.
Cf. A088690, A380828.
Cf. A376094.

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

E.g.f. A(x) satisfies A(x) = exp(3*x*A(x)) / ( 1 - x*exp(3*x*A(x)) ).
a(n) = n! * Sum_{k=0..n} (3*n+3)^k * binomial(n,k)/(k+1)!.
a(n) = A376094(n+1)/(n+1).

A361212 E.g.f. satisfies A(x) = exp( 3xA(x) / (1-x) ).

Original entry on oeis.org

1, 3, 33, 612, 16353, 576108, 25306803, 1334701854, 82258866225, 5805344935368, 461848917299499, 40904277651802458, 3992219566916292873, 425766991650939828828, 49266876888419716251315, 6147944525591645916094182, 823045511075200872642258273

Offset: 0

Seiichi Manyama, Mar 04 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..327
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A360939.

Cf. A361066, A361182.

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

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

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

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

A380829 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3x) / (1 + xexp(-x)) ).

Original entry on oeis.org

1, 4, 45, 891, 25757, 986653, 47235873, 2718521725, 182963698521, 14107443728553, 1226582182222469, 118751669770995913, 12671598073554789909, 1477709279563430592877, 186988047586389278202633, 25518989446806209718773157, 3736444151435292273253963313, 584269287631534621583659461841

Offset: 0

Seiichi Manyama, Feb 05 2025

nonn

Index entries for reversions of series

Cf. A361182, A380826, A380830.

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

E.g.f. A(x) satisfies A(x) = exp(3*x*A(x)) / ( 1 - x*exp(2*x*A(x)) ).

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

cp's OEIS Frontend

A360609 E.g.f. satisfies A(x) = exp(x*A(x)^3) / (1-x).

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A361193 E.g.f. satisfies A(x) = exp( -2*x*A(x) ) / (1-x).

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A361194 E.g.f. satisfies A(x) = exp( -3*x*A(x) ) / (1-x).

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A380826 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x*exp(-2*x)) ).

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A380830 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x) ).

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A361212 E.g.f. satisfies A(x) = exp( 3*x*A(x) / (1-x) ).

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PARI

PARI

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A380829 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3*x) / (1 + x*exp(-x)) ).

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PARI

Formula

A361193 E.g.f. satisfies A(x) = exp( -2xA(x) ) / (1-x).

A361194 E.g.f. satisfies A(x) = exp( -3xA(x) ) / (1-x).

A380826 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3x) / (1 + xexp(-2*x)) ).

A361212 E.g.f. satisfies A(x) = exp( 3xA(x) / (1-x) ).

A380829 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-3x) / (1 + xexp(-x)) ).