A052868 -id:A052868 - cp's OEIS frontend

A362775 E.g.f. satisfies A(x) = exp( x/(1-x)^2 * A(x) ).

1, 1, 7, 70, 965, 17216, 379207, 9969772, 305154313, 10668593008, 419714689931, 18358646058644, 884070662867053, 46486344447041032, 2650567497877525423, 162908800485532424236, 10737607698626311094033, 755571950776792829919968

Offset: 0

Seiichi Manyama, May 02 2023

nonn

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

Cf. A082579, A362776.

Cf. A052868, A361065.

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

E.g.f.: exp( -LambertW(-x/(1-x)^2) ).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+k-1,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: -LambertW(-x/(1-x)^2) * (1-x)^2 / x.
a(n) ~ 2^(n + 1/2) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * n^(n-1) / ((-1 + sqrt(1 + 4*exp(-1)))^(3/2) * (1 + 2*exp(-1) - sqrt(1 + 4*exp(-1)))^(n - 1/2) * exp(2*n-1)). (End)

A335945 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 + x)).

Original entry on oeis.org

1, 1, 1, 4, 17, 116, 907, 9010, 102097, 1348408, 19939571, 330204854, 6015657529, 120016789348, 2597201945899, 60667591974826, 1520434054966433, 40710815980598000, 1159627208850209251, 35018022339726428926, 1117395892399939407241, 37569709612314269554396

Offset: 0

Ilya Gutkovskiy, Jul 01 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A060356, A108919.

nmax = 21; A[] = 0; Do[A[x] = Exp[x A[x]/(1 + x)] + O[x]^(nmax + 1) // Normal, nmax + 1];CoefficientList[A[x], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[-(1 + x) LambertW[-x/(1 + x)]/x, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) Binomial[n - 1, k - 1] (k + 1)^(k - 1) n!/k!, {k, 0, n}], {n, 0, 21}]

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1+x))))) \\ Seiichi Manyama, Mar 05 2023

E.g.f.: -(1 + x) * LambertW(-x/(1 + x)) / x.
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n-1,k-1) * (k+1)^(k-1) * n! / k!.
a(n) ~ (exp(1) - 1)^(n + 1/2) * n^(n-1) / exp(n - 1/2). - Vaclav Kotesovec, Jul 01 2020
E.g.f.: exp ( -LambertW(-x/(1+x)) ). - Seiichi Manyama, Mar 05 2023

A370436 Expansion of e.g.f. A(x) satisfying A(x) = Product_{n>=1} cosh(x^n*A(x)).

Original entry on oeis.org

1, 1, 25, 1801, 251665, 60023281, 21783217897, 11244708818617, 7836581579364769, 7098342754565616481, 8108890187934052712761, 11407792409056590722072041, 19382875328830017602572089265, 39137558151287789768856541250641, 92642186419738783260791047129751305

Offset: 0

Paul D. Hanna, Feb 29 2024

nonn

A related identity is sinh(x)/x = Product_{n>=1} cosh(x/2^n).
Motivated by the fixed point sr = Product_{n>=1} cosh(sr*(1/2)^n), where sr = sinh(sr)/sr = 1.3132837183534835944... (A133916).
The radius of convergence of e.g.f. A(x) is r = 0.536180023946684527...; A(x) evaluated at x = r satisfies (1) A(r) = 1 / [Sum_{n>=1} r^n * tanh(r^n * A(r))] and (2) A(r) = Product_{n>=1} cosh(r^n * A(r)), where A(r) = 1.726459616031644143... Note that r = 2/sqrt(d) where d is given in Vaclav Kotesovec's formula. - Paul D. Hanna, Mar 01 2024

			E.g.f.: A(x) = 1 + x^2/2! + 25*x^4/4! + 1801*x^6/6! + 251665*x^8/8! + 60023281*x^10/10! + 21783217897*x^12/12! + 11244708818617*x^14/14! + ...
where
A(x) = cosh(x*A(x)) * cosh(x^2*A(x)) * cosh(x^3*A(x)) * cosh(x^4*A(x)) * ...
RELATED SERIES.
log(A(x)) = x^2/2! + 22*x^4/4! + 1456*x^6/6! + 189232*x^8/8! + 43031296*x^10/10! + 15070050304*x^12/12! + 7582799641600*x^14/14! + ...
where the logarithm of A(x) may be written as
log(A(x)) = A(x)^2*x^2/((1-x^2)*2!) - 2*A(x)^4*x^4/((1-x^4)*4!) + 16*A(x)^6*x^6/((1-x^6)*6!) - 272*A(x)^8*x^8/((1-x^8)*8!) +- ...
in which the coefficients (A000182) are taken from the series for
log(cosh(x)) = x^2/2! - 2*x^4/4! + 16*x^6/6! - 272*x^8/8! + 7936*x^10/10! - 353792*x^12/12! + ... + (-1)^(n-1)*A000182(n)*x^(2*n)/(2*n)! + ...
SPECIFIC VALUES.
A(1/2) = sr = sinh(sr)/sr = 1.3132837183534835944... (A133916).
A(1/3) = 1.0732039012742053466040583737125980229145749599...
A(1/4) = 1.0360440144515230397124814819872716708635571534...
A(1/5) = 1.0218446382532843162980010372360052478618874316...
A(1/6) = 1.0147502698567245499165367962927287752921277594...
A(1/8) = 1.0080767433235165071886167844759835649060174961...

Paul D. Hanna, Table of n, a(n) for n = 0..201

Cf. A052868, A370339, A133916.

nmax = 20; A[] = 0; Do[A[x] = Product[Cosh[x^k*A[x]], {k, 1, 2*nmax}] + O[x]^(2*nmax + 1) // Normal, 2*nmax + 1]; Table[(CoefficientList[A[x], x]*Range[0, 2*nmax]!)[[2*j-1]], {j, 1, nmax}] (* Vaclav Kotesovec, Mar 01 2024 *)

{a(n) = my(A=1); for(m=1, n+1, A=truncate(A);
A = prod(k=1, m, cosh(x^k*A +O(x^(2*m+1))) ) ; ); (2*n)!*polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! satisfies the following formulas.
(1) A(x) = Product_{n>=1} cosh(x^n*A(x)).
(2) log(A(x)) = Sum_{n>=1} (-1)^(n-1)*A000182(n) * A(x)^(2*n) * x^(2*n)/((1-x^(2*n))*(2*n)!), where A000182 are the tangent numbers.
a(n) ~ c * d^n * (n-1)!^2, where d = 13.91357494878057309372... and c = 0.2556213529084458598... - Vaclav Kotesovec, Mar 01 2024

A360939 E.g.f. satisfies A(x) = exp( 2xA(x) / (1-x) ).

Original entry on oeis.org

1, 2, 16, 212, 4016, 99952, 3096448, 115063328, 4993598464, 248071645952, 13888585800704, 865481914527232, 59426130052458496, 4458258196636276736, 362864617248019800064, 31848507841521274769408, 2998685833332127139299328, 301504120063370711801724928

Offset: 0

Seiichi Manyama, Mar 04 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..342
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361212.

Cf. A361065.

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

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

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

a(n) = n! * Sum_{k=0..n} 2^k * (k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp ( -LambertW(-2*x/(1-x)) ).
E.g.f.: -(1-x)/(2*x) * LambertW(-2*x/(1-x)).
a(n) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (sqrt(2) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 10 2023

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)).

A367790 E.g.f. satisfies A(x) = exp( x/(1-x)^4 * A(x) ).

Original entry on oeis.org

1, 1, 11, 148, 2669, 62056, 1777927, 60692920, 2408692505, 109074596320, 5553702114731, 314208715035304, 19561795753879909, 1329317730339826384, 97924919301787209647, 7773978186375852940696, 661702605336795904770353, 60119367618216155944350400

Offset: 0

Seiichi Manyama, Nov 30 2023

nonn

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

Cf. A052868, A362775, A367789.

Cf. A162475, A361283.

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(n+3*k-1,n-k)/k!.

A376494 E.g.f. satisfies A(x) = exp(x^2 * A(x)^2 / (1 - x)).

Original entry on oeis.org

1, 0, 2, 6, 84, 720, 12000, 178920, 3744720, 79531200, 2056652640, 56284351200, 1753673423040, 58443081016320, 2142625074670080, 83948606126985600, 3549356731374854400, 159643527455123712000, 7656564912324122995200

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

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

Cf. A052868, A376495.

Cf. A052845, A376474.

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(n-k-1, 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(n-k-1,n-2*k)/k!.
a(n) ~ sqrt(16 + 2*exp(-1) - 2*exp(-1/2)*sqrt(exp(-1)+8)) * (exp(1/2)*sqrt(exp(-1)+8) - 1) * 2^(2*n-2) * n^(n-1) / ((4 + exp(-1) - exp(-1/2)*sqrt(exp(-1)+8)) * (sqrt(1 + 8*exp(1)) - 1)^n). - Vaclav Kotesovec, Aug 05 2025

A376495 E.g.f. satisfies A(x) = exp(x^3 * A(x)^3 / (1 - x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 3240, 40320, 463680, 11491200, 248572800, 4869849600, 135896745600, 4017466252800, 113150157120000, 3765622699238400, 137549036072448000, 5019223860338688000, 199794776937044889600, 8636618647667288678400

Offset: 0

Seiichi Manyama, Sep 25 2024

nonn

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

Cf. A052868, A376494.

Cf. A376475.

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(n-2*k-1, 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(n-2*k-1,n-3*k)/k!.

A372158 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) / (1 - x) ).

Original entry on oeis.org

1, 2, 12, 110, 1368, 21602, 415036, 9416094, 246730448, 7340456258, 244615296564, 9030708939518, 365998814372824, 16159576541122146, 772216069907880812, 39715949460883093598, 2187682975276318552224, 128508919233259720967810

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

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

Cf. A052868, A372159.

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(n+(s-1)*k-1, 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(n+(s-1)*k-1,n-k)/k!.
a(n) ~ 2 * exp(2) * (1 + exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Aug 05 2025

A372159 E.g.f. A(x) satisfies A(x) = exp( 3 * x * A(x)^(1/3) / (1 - x) ).

Original entry on oeis.org

1, 3, 21, 216, 2937, 49788, 1013247, 24106134, 657277185, 20225122632, 693755934159, 26261393088978, 1087866116802081, 48965716033901436, 2380245527593532559, 124300353332797939422, 6941285402232405794817, 412817223292008085699344

Offset: 0

Seiichi Manyama, Apr 20 2024

nonn

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

Cf. A052868, A372158.

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(n+(s-1)*k-1, 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(n+(s-1)*k-1,n-k)/k!.

cp's OEIS Frontend

A362775 E.g.f. satisfies A(x) = exp( x/(1-x)^2 * A(x) ).

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A335945 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 + x)).

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A370436 Expansion of e.g.f. A(x) satisfying A(x) = Product_{n>=1} cosh(x^n*A(x)).

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A360939 E.g.f. satisfies A(x) = exp( 2*x*A(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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A367790 E.g.f. satisfies A(x) = exp( x/(1-x)^4 * A(x) ).

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A376494 E.g.f. satisfies A(x) = exp(x^2 * A(x)^2 / (1 - x)).

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A376495 E.g.f. satisfies A(x) = exp(x^3 * A(x)^3 / (1 - x)).

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A360939 E.g.f. satisfies A(x) = exp( 2xA(x) / (1-x) ).

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