A352410 -id:A352410 - cp's OEIS frontend

A352412 E.g.f.: 2x / LambertW( 2x/(1-x) ).

1, 1, -4, 20, -224, 3392, -67232, 1629728, -46799104, 1552143104, -58386807296, 2455954797056, -114222622662656, 5819845970653184, -322384671892123648, 19290013218140254208, -1239886482366130946048, 85200320552417960394752

Offset: 0

Paul D. Hanna, Mar 15 2022

sign

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

			E.g.f.: A(x) = 1 + x - 4*x^2/2! + 20*x^3/3! - 224*x^4/4! + 3392*x^5/5! - 67232*x^6/6! + 1629728*x^7/7! - 46799104*x^8/8! + ...
such that A(x) = (1-x) * exp(2*x/A(x)), where
exp(2*x/A(x)) = 1 + 2*x + 20*x^3/3! - 144*x^4/4! + 2672*x^5/5! - 51200*x^6/6! + 1271328*x^7/7! - 36628480*x^8/8! + ...
Related series.
The e.g.f. A(x) satisfies A( x/(exp(-2*x) + x) ) = 1/(exp(-2*x) + x), where
1/(exp(-2*x) + x) = 1 + x - 2*x^2/2! - 10*x^3/3! + 24*x^4/4! + 312*x^5/5! - 560*x^6/6! + ... + A336958(n)*(-x)^n/n! + ...
Related table.
Another defining property of the e.g.f. A(x) is illustrated here.
The table of coefficients of x^k/k! in A(x)^n begins:
n=1: [1, 1, -4,   20,  -224, 3392, -67232, 1629728, ...];
n=2: [1, 2, -6,   16,  -192, 2944, -58880, 1434752, ...];
n=3: [1, 3, -6,   -6,   -48, 1296, -29664,  776544, ...];
n=4: [1, 4, -4,  -40,    88,  128,  -7424,  263936, ...];
n=5: [1, 5,  0,  -80,   120,  280,   -320,   38720, ...];
n=6: [1, 6,  6, -120,   -24, 1872,  -3312,     768, ...];
n=7: [1, 7, 14, -154,  -392, 4424,  -3920,  -22288, ...];
...
from which we can illustrate that the partial sum of coefficients of x^k, k=0..n, in A(x)^n equals zero, for n > 1, as follows:
n=1: 2 = 1 + 1;
n=2: 0 = 1 + 2 + -6/2!;
n=3: 0 = 1 + 3 + -6/2! +   -6/3!;
n=4: 0 = 1 + 4 + -4/2! +  -40/3! +   88/4!;
n=5: 0 = 1 + 5 +  0/2! +  -80/3! +  120/4! +  280/5!;
n=6: 0 = 1 + 6 +  6/2! + -120/3! +  -24/4! + 1872/5! + -3312/6!;
n=7: 0 = 1 + 7 + 14/2! + -154/3! + -392/4! + 4424/5! + -3920/6! + -22288/7!;
...

Cf. A352410, A352411, A352448, A336958.

{a(n) = n!*polcoeff( 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(2*x/lambertw(2*x/(1-x)))) \\ Michel Marcus, Mar 17 2022

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies:
(1) A(x) = 2*x / LambertW( 2*x/(1-x) ).
(2) A(x) = (1-x) * exp( 2*x/A(x) ).
(3) A(x) = 2*x / log( A(x)/(1-x) ).
(4) A( x/(exp(-2*x) + x) ) = 1/(exp(-2*x) + x).
(5) A(x) = x / Series_Reversion( x/(exp(-2*x) + x) ).
(6) Sum_{k=0..n} [x^k] A(x)^n = 0, for n > 1.
(7) [x^(n+1)/(n+1)!] A(x)^n = -(-2)^(n+1) * n for n >= (-1).
a(n) ~ (-1)^(n+1) * exp(-1) * sqrt(2) * (2 - exp(-1))^(n - 1/2) * n^(n-1). - Vaclav Kotesovec, Mar 15 2022

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).

A370875 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^2)) ).

Original entry on oeis.org

1, 1, 4, 24, 228, 2820, 44400, 840000, 18669840, 475871760, 13698296640, 439402803840, 15545690233920, 601352177025600, 25251437978807040, 1143932660001331200, 55612090342967558400, 2887929114414030086400, 159548423949650274739200

Offset: 0

Seiichi Manyama, Mar 03 2024

nonn

Index entries for reversions of series

Cf. A352410, A370876.

Cf. A360601.

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

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

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

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

A370876 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^3)) ).

Original entry on oeis.org

1, 1, 2, 12, 120, 1320, 17640, 304920, 6249600, 143579520, 3711052800, 107762054400, 3455138332800, 120802387305600, 4583177081683200, 187766031131078400, 8256125218115174400, 387662886088250572800, 19364540503274942976000, 1025507260911983244595200

Offset: 0

Seiichi Manyama, Mar 03 2024

nonn

Index entries for reversions of series

Cf. A352410, A370875.

Cf. A360609.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(x+exp(x^3)))/x))

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

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

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

A371038 E.g.f. satisfies A(x) = exp(x^2*A(x)) / (1-x).

Original entry on oeis.org

1, 1, 4, 18, 132, 1140, 12720, 164640, 2514960, 43500240, 850076640, 18418609440, 439831909440, 11457415569600, 323707663319040, 9854548934630400, 321709145793235200, 11209975693710393600, 415330670608805952000, 16303720885477254028800

Offset: 0

Seiichi Manyama, Mar 09 2024

nonn

Robert Israel, Table of n, a(n) for n = 0..399
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A352410, A371039.

E:=  LambertW( -x^2/(1-x) ) / (-x^2):
S:= series(E,x,43):
seq(coeff(S,x,i)*i!,i=0..40); # Robert Israel, Mar 09 2025

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

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

E.g.f.: LambertW( -x^2/(1-x) ) / (-x^2).
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)^(k-1) * binomial(n-k,n-2*k)/k!.
a(n) ~ exp(2) * sqrt(1 + 4*exp(1) - sqrt(1 + 4*exp(1))) * 2^(n + 3/2) * n^(n-1) / ((1 + 2*exp(1) - sqrt(1 + 4*exp(1)))*(-1 + sqrt(1 + 4*exp(1)))^(n+1)). - Vaclav Kotesovec, Mar 12 2024

A377888 E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)^2).

Original entry on oeis.org

1, 2, 17, 289, 7541, 267041, 11974645, 650666731, 41560476809, 3052145052433, 253400719220801, 23470964805942083, 2399562226994185885, 268404500411311273465, 32606551238103342068717, 4275233840499570086190331, 601753408713140793660643985, 90500525005651471292191270433

Offset: 0

Seiichi Manyama, Nov 10 2024

nonn

Cf. A352410, A377831, A377889.

Cf. A371318.

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

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

A377889 E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)^3).

Original entry on oeis.org

1, 2, 21, 472, 16581, 795736, 48509641, 3589729760, 312603962985, 31321633489408, 3549706188092541, 448973808123051520, 62697159481460439469, 9581292408000225087488, 1590488540940006100524657, 284993765391981838318575616, 54826610288277007690469896017

Offset: 0

Seiichi Manyama, Nov 10 2024

nonn

Cf. A352410, A377831, A377888.

Cf. A373324.

a(n) = n!*sum(k=0, n, (3*n-2*k+1)^(k-1)*binomial(4*n-3*k, n-k)/k!);

a(n) = n! * Sum_{k=0..n} (3*n-2*k+1)^(k-1) * binomial(4*n-3*k,n-k)/k!.

A370877 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^2/2)) ).

Original entry on oeis.org

1, 1, 3, 15, 111, 1095, 13605, 204225, 3597825, 72788625, 1663323795, 42373980495, 1190822561775, 36596898673335, 1221033470181525, 43954996792932225, 1698138394110583425, 70082689941923083425, 3077205709746516423075, 143235112906380591471375

Offset: 0

Seiichi Manyama, Mar 03 2024

nonn

Index entries for reversions of series

Cf. A352410, A370878.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(x+exp(x^2/2)))/x))

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

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

A370878 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^3/6)) ).

Original entry on oeis.org

1, 1, 2, 7, 40, 320, 3190, 37870, 526400, 8434720, 153092800, 3099958400, 69237737800, 1691184094600, 44855672061200, 1283910696468400, 39445370739174400, 1294688750568012800, 45213628130719048000, 1673957726914620347200, 65493419262155812928000

Offset: 0

Seiichi Manyama, Mar 03 2024

nonn

Index entries for reversions of series

Cf. A352410, A370877.

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x/(x+exp(x^3/6)))/x))

a(n) = n!*sum(k=0, n\3, (3*k+1)^(k-1)*binomial(n, 3*k)/(6^k*k!));

a(n) = n! * Sum_{k=0..floor(n/3)} (3*k+1)^(k-1) * binomial(n,3*k)/(6^k * k!).

cp's OEIS Frontend

A352412 E.g.f.: 2*x / LambertW( 2*x/(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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A370875 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^2)) ).

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A370876 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^3)) ).

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A371038 E.g.f. satisfies A(x) = exp(x^2*A(x)) / (1-x).

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A377888 E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)^2).

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A377889 E.g.f. A(x) satisfies A(x) = exp(x * A(x))/(1 - x*A(x)^3).

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A370877 Expansion of e.g.f. (1/x) * Series_Reversion( x/(x + exp(x^2/2)) ).

A352412 E.g.f.: 2x / LambertW( 2x/(1-x) ).

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

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