A361068 -id:A361068 - cp's OEIS frontend

A352448 Expansion of e.g.f. LambertW( -2x/(1-x) ) / (-2x).

1, 3, 22, 278, 5128, 125592, 3850000, 142013328, 6129705088, 303238991744, 16920975718144, 1051612647426816, 72045481821580288, 5394849460316820480, 438392509692455286784, 38424395486908104071168, 3613476161122656804438016

Offset: 0

Paul D. Hanna, Mar 16 2022

nonn

An interesting property of this e.g.f. A(x) is that the sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1.

			E.g.f.: A(x) = 1 + 3*x + 22*x^2/2! + 278*x^3/3! + 5128*x^4/4! + 125592*x^5/5! + 3850000*x^6/6! + 142013328*x^7/7! + ...
such that A(x) = exp( 2*x*A(x) ) / (1-x), where
exp( 2*x*A(x) ) = 1 + 2*x + 16*x^2/2! + 212*x^3/3! + 4016*x^4/4! + 99952*x^5/5! + 3096448*x^6/6! + 115063328*x^7/7! + ...
Related table.
Another interesting property of the e.g.f. A(x) is illustrated here.
The table of coefficients of x^k/k! in 1/A(x)^n begins:
n=1: [1,  -3,  -4,   -44,  -736,  -16832, -491168, ...];
n=2: [1,  -6,  10,   -16,  -320,   -8064, -249344, ...];
n=3: [1,  -9,  42,   -78,   -48,   -1776,  -66528, ...];
n=4: [1, -12,  92,  -392,   728,    -128,   -8960, ...];
n=5: [1, -15, 160, -1120,  4600,   -8520,    -320, ...];
n=6: [1, -18, 246, -2424, 16104,  -64752,  119952, ...];
...
from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in 1/A(x)^n equals zero, for n > 1, as follows:
n=1:-2 = 1 +  -3;
n=2: 0 = 1 +  -6 +  10/2!;
n=3: 0 = 1 +  -9 +  42/2! +   -78/3!;
n=4: 0 = 1 + -12 +  92/2! +  -392/3! +   728/4!;
n=5: 0 = 1 + -15 + 160/2! + -1120/3! +  4600/4! +   -8520/5!;
n=6: 0 = 1 + -18 + 246/2! + -2424/3! + 16104/4! +  -64752/5! +  119952/6!;
...

Seiichi Manyama, Table of n, a(n) for n = 0..344

Cf. A352410, A352411, A352412.

Cf. A361068.

terms = 17; A[] = 0; Do[A[x] = Exp[2x*A[x]]/(1-x) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Mar 24 2025 *)

{a(n) = n!*polcoeff( (1/x)*serreverse( x/(exp(2*x  +x^2*O(x^n)) + x) ),n)}
for(n=0,30,print1(a(n),", "))

my(x='x+O('x^30)); Vec(serlaplace(lambertw(-2*x/(1-x))/(-2*x))) \\ Michel Marcus, Mar 17 2022

a(n) = n!*sum(k=0, n, 2^k*(k+1)^(k-1)*binomial(n, k)/k!); \\ Seiichi Manyama, Mar 03 2023

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = LambertW( -2*x/(1-x) ) / (-2*x).
(2) A(x) = exp( 2*x*A(x) ) / (1-x).
(3) A(x) = log( (1-x) * A(x) ) / (2*x).
(4) A( x/(exp(2*x) + x) ) = exp(2*x) + x.
(5) A(x) = (1/x) * Series_Reversion( x/(exp(2*x) + x) ).
(6) Sum_{k=0..n} [x^k] 1/A(x)^n = 0, for n > 1.
(7) [x^(n+1)/(n+1)!] 1/A(x)^n = -2^(n+1) * n for n >= (-1).
a(n) ~ (1 + 2*exp(1))^(n + 3/2) * n^(n-1) / (2^(3/2) * exp(n + 1/2)). - Vaclav Kotesovec, Mar 18 2022
a(n) = n! * Sum_{k=0..n} 2^k * (k+1)^(k-1) * binomial(n,k)/k!. - Seiichi Manyama, Mar 03 2023

A361065 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).

Original entry on oeis.org

1, 1, 7, 85, 1521, 36421, 1097743, 39968601, 1707558401, 83777885929, 4643185678551, 286930307457949, 19562851003118833, 1458832806486727725, 118121195050068075167, 10320576944751955718881, 967863775658734350214017, 96970880819175875321264209

Offset: 0

Seiichi Manyama, Mar 01 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, A361066, A361067, A361068, A361069.

A361065 := proc(n)
    add((2*k+1)^(k-1)*binomial(n-1,n-k)/k!,k=0..n) ;
    %*n! ;
end proc:
seq(A361065(n),n=0..10) ; # R. J. Mathar, Mar 02 2023

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^2] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

a(n) = n!*sum(k=0, n, (2*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))/2)))

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

a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( -LambertW(-2*x/(1-x))/2 ).
E.g.f.: sqrt( -(1-x)/(2*x) * LambertW(-2*x/(1-x)) ).
a(n) ~ (1 + 2*exp(1))^(n + 1/2) * n^(n-1) / (2^(3/2) * exp(n)). - Vaclav Kotesovec, Mar 02 2023

A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

Original entry on oeis.org

1, 1, 9, 148, 3673, 123276, 5234599, 269262022, 16279709793, 1131627199816, 88926737901031, 7796168316687906, 754414052156289265, 79872584117422215484, 9184299004593618881655, 1139822558262829096519726, 151857077047173825979147969

Offset: 0

Seiichi Manyama, Mar 01 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, A361065, A361067, A361068, A361069.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

a(n) = n!*sum(k=0, n, (3*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))/3)))

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

a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( -LambertW(-3*x/(1-x))/3 ).
E.g.f.: ( -(1-x)/(3*x) * LambertW(-3*x/(1-x)) )^(1/3).
a(n) ~ (1 + 3*exp(1))^(n + 1/2) * n^(n-1) / (3^(3/2) * exp(n + 1/6)). - Vaclav Kotesovec, Mar 02 2023

A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

Original entry on oeis.org

1, 1, -3, 40, -719, 18396, -598157, 23713726, -1108701519, 59735988424, -3644505746549, 248358786667674, -18697767289462967, 1541202721786228060, -138046868771541971373, 13351368704222195975206, -1386710317839048140282783, 153939247458296219191539984

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

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

Cf. A052868, A361065, A361066, A361067, A361068.

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x]^3)] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

a(n) = n!*sum(k=0, n, (-3*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))/3)))

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

a(n) = n! * Sum_{k=0..n} (-3*k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(3*x/(1-x))/3 ).
E.g.f.: 1 / ( (1-x)/(3*x) * LambertW(3*x/(1-x)) )^(1/3).
a(n) ~ (-1)^(n+1) * 3^(-3/2) * exp(-1/3) * (3 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024

A361067 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)) ).

Original entry on oeis.org

1, 1, 1, 4, 9, 76, 175, 3606, 833, 354376, -1605249, 65111410, -718371071, 20105327100, -351241054177, 9362931464446, -214514949732735, 6039303900168976, -165679758877120001, 5093296357218337386, -159900268661169533119, 5405435526807425433220

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..413
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A052868, A361065, A361066, A361068, A361069.

nmax = 21; A[_] = 1;
Do[A[x_] = Exp[x/((1 - x)*A[x])] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

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

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

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

a(n) = n! * Sum_{k=0..n} (-k+1)^(k-1) * binomial(n-1,n-k)/k!.
E.g.f.: exp( LambertW(x/(1-x)) ).
E.g.f.: 1 / ( (1-x)/x * LambertW(x/(1-x)) ).
a(n) ~ -(-1)^n * exp(-1) * (1 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 02 2023

A361091 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 1, -71, -19, 10051, 12349, -3185391, -9346247, 1797304771, 9717361721, -1582301193527, -13722004186331, 2000705907453891, 25552516703201461, -3432004488804778079, -60960914621687232271, 7660860906885122096515

Offset: 0

Seiichi Manyama, Mar 01 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..380

Cf. A161630, A212722, A212917, A361090, A361092.

Cf. A361068, A361096.

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

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

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).

A361213 E.g.f. satisfies A(x) = exp( 2xA(x) / (1+x) ).

Original entry on oeis.org

1, 2, 8, 68, 848, 14192, 298048, 7546016, 223792640, 7612381952, 292216807424, 12497875215872, 589392367925248, 30386736933804032, 1700376343771136000, 102641314849948602368, 6648428846464054919168, 459977466799800897437696

Offset: 0

Seiichi Manyama, Mar 04 2023

nonn

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

Cf. A335945, A361214.

Cf. A361068, A361193.

a(n) = (-1)^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) = (-1)^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) ~ (2*exp(1) - 1)^(n + 1/2) * n^(n-1) / (sqrt(2) * exp(n - 1/2)). - Vaclav Kotesovec, Nov 10 2023

A365039 E.g.f. satisfies A(x) = exp(x * (1 + x)/A(x)^2).

Original entry on oeis.org

1, 1, -1, 7, -79, 1201, -22961, 530167, -14372191, 447825889, -15776617249, 620209389031, -26918670325295, 1278598424153233, -65973615445792081, 3674793950748867031, -219773335672937703871, 14046128883828030510529, -955409650156763223984449

Offset: 0

Seiichi Manyama, Aug 18 2023

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

Cf. A362771, A362773, A363478, A365038, A365040.

Cf. A361068.

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

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

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

cp's OEIS Frontend

A352448 Expansion of e.g.f. LambertW( -2*x/(1-x) ) / (-2*x).

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A361065 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^2 ).

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A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

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A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

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A361067 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)) ).

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A361091 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^2) ).

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

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A352448 Expansion of e.g.f. LambertW( -2x/(1-x) ) / (-2x).

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

A361213 E.g.f. satisfies A(x) = exp( 2xA(x) / (1+x) ).