A048802 -id:A048802 - cp's OEIS frontend

A357346 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(A(x)).

0, 1, 5, 52, 849, 18996, 540986, 18726247, 763480675, 35837071558, 1903538106065, 112880374866172, 7392418912962210, 529898419942327801, 41266682731537698181, 3469461853041348996044, 313200848521114144611273, 30215925892728362737156556

Offset: 0

Seiichi Manyama, Sep 25 2022

nonn

Index entries for reversions of series

Cf. A052888, A357347, A357348, A357424.

Cf. A048802, A349557.

a(n) = sum(k=1, n, (n+k)^(k-1)*stirling(n, k, 2));

a(n) = Sum_{k=1..n} (n+k)^(k-1) * Stirling2(n,k).

E.g.f.: Series_Reversion( exp(-x) * log(1 + x * exp(-x)) ). - Seiichi Manyama, Sep 09 2024

A355179 Expansion of e.g.f. -LambertW(x^2 * (1 - exp(x)))/2.

Original entry on oeis.org

0, 0, 0, 3, 6, 10, 375, 2541, 11788, 317556, 4238685, 37921015, 909616026, 18283276518, 261259582675, 6360432558585, 164704011195480, 3332419310132776, 88606184592031353, 2713050497589230763, 71412977041725823750, 2144089948615678382970

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A048802, A355180, A357267.

With[{nn=30},CoefficientList[Series[(-LambertW[x^2 (1-Exp[x])])/2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 07 2025 *)

my(N=20, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2*(1-exp(x)))/2)))

a(n) = n!*sum(k=1, n\3, k^(k-1)*stirling(n-2*k, k, 2)/(n-2*k)!)/2;

a(n) = (n!/2) * Sum_{k=1..floor(n/3)} k^(k-1) * Stirling2(n-2*k,k)/(n-2*k)!.

A355180 Expansion of e.g.f. -LambertW(x^3 * (1 - exp(x)))/6.

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 20, 35, 6776, 60564, 352920, 1663365, 126625180, 2361079006, 27334747804, 245495250455, 11174333090480, 328952158255400, 6245314009946736, 90576650639967369, 3209305759254634740, 122557203047084965810, 3365068665450300234580

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A048802, A355179, A357267.

my(N=20, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3*(1-exp(x)))/6)))

a(n) = n!*sum(k=1, n\4, k^(k-1)*stirling(n-3*k, k, 2)/(n-3*k)!)/6;

a(n) = (n!/6) * Sum_{k=1..floor(n/4)} k^(k-1) * Stirling2(n-3*k,k)/(n-3*k)!.

A356000 Expansion of e.g.f. -LambertW((1 - exp(2*x))/2).

Original entry on oeis.org

0, 1, 4, 25, 236, 3061, 50670, 1020881, 24245576, 663290281, 20541118266, 710366714773, 27135242829436, 1134708855427629, 51556563327940390, 2529164265815033241, 133229047747850647312, 7500633471737652798673, 449445732625670948076530

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.
Index entries for reversions of series

Cf. A048802, A356001.

Cf. A357335.

With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[(1 - Exp[2*x])/2], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw((1-exp(2*x))/2))))

a(n) = sum(k=1, n, 2^(n-k)*k^(k-1)*stirling(n, k, 2));

a(n) = Sum_{k=1..n} 2^(n-k) * k^(k-1) * Stirling2(n,k).
a(n) ~ 2^(n - 1/2) * sqrt(exp(1) + 2) * n^(n-1) / (exp(n) * (log(exp(1) + 2) - 1)^(n - 1/2)). - Vaclav Kotesovec, Oct 04 2022
E.g.f.: Series_Reversion( (log(1 + 2 * x * exp(-x)))/2 ). - Seiichi Manyama, Sep 11 2024

A356001 Expansion of e.g.f. -LambertW((1 - exp(3*x))/3).

Original entry on oeis.org

0, 1, 5, 36, 379, 5461, 100476, 2250613, 59432141, 1807959042, 62262816157, 2394551966401, 101724440338494, 4730814590128615, 239057921691911861, 13042779411190737420, 764136645388807739239, 47846833035272035228849, 3188740106752561252031364

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.
Index entries for reversions of series

Cf. A048802, A356000.

Cf. A357336.

With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[(1 - Exp[3*x])/3], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw((1-exp(3*x))/3))))

a(n) = sum(k=1, n, 3^(n-k)*k^(k-1)*stirling(n, k, 2));

a(n) = Sum_{k=1..n} 3^(n-k) * k^(k-1) * Stirling2(n,k).
a(n) ~ 3^(n - 1/2) * sqrt(exp(1) + 3) * n^(n-1) / (exp(n) * (log(exp(1) + 3) - 1)^(n - 1/2)). - Vaclav Kotesovec, Oct 04 2022
E.g.f.: Series_Reversion( (log(1 + 3 * x * exp(-x)))/3 ). - Seiichi Manyama, Sep 11 2024

A216858 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} summed over all subsets.

Original entry on oeis.org

0, 1, 5, 38, 422, 6184, 112632, 2453296, 62202800, 1799623296, 58507176320, 2111633645824, 83777729991936, 3624054557443072, 169759643117603840, 8560585769442662400, 462387289560368764928, 26633435981686107701248, 1629609677806398679646208, 105555926477075661655441408, 7215930505311133152120995840

Offset: 0

Geoffrey Critzer, Sep 17 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 0..370

Cf. A216857, A000248, A048802, A072034.

nn=20; a=-ProductLog[-x Exp[x] ]; Range[0,nn]! CoefficientList[Series[Log[1/(1-a)], {x,0,nn}], x]

x='x+O('x^30); concat([0], Vec(serlaplace(log(1/(1+ lambertw(-x*exp(x))))))) \\ G. C. Greubel, Nov 16 2017

E.g.f.: log(1/(1-T(x*exp(x)))) where T(x) is the e.g.f. for A000169.

a(n) ~ n!/(2*n*LambertW(exp(-1))^n) * (1 - sqrt(2*(1 + LambertW(exp(-1))) /(Pi*n))/3). - Vaclav Kotesovec, Sep 24 2013

A355181 Expansion of e.g.f. -LambertW(x^2/2 * (1 - exp(x))).

Original entry on oeis.org

0, 0, 0, 3, 6, 10, 195, 1281, 5908, 90756, 1098765, 9605035, 147947646, 2496239538, 33836915203, 588360763095, 12104789358600, 223722576473896, 4578806487368313, 108875473376842467, 2519418390663035170, 60831875074927797750, 1640260621340460494991

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A048802, A355308, A357267.

my(N=30, x='x+O('x^N)); concat([0, 0, 0], Vec(serlaplace(-lambertw(x^2/2*(1-exp(x))))))

a(n) = n!*sum(k=1, n\3, k^(k-1)*stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));

a(n) = n! * Sum_{k=1..floor(n/3)} k^(k-1) * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).

A355308 Expansion of e.g.f. -LambertW(x^3/6 * (1 - exp(x))).

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 20, 35, 1176, 10164, 58920, 277365, 4472380, 69189406, 772011604, 6861855455, 95279504880, 1819310613800, 30768119885136, 430200439251369, 6770486332450740, 139958614722287410, 3033142442978720380, 58782387380290683571, 1138026666874389737544

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A048802, A355181, A357267.

Cf. A353999, A355180.

my(N=30, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(serlaplace(-lambertw(x^3/6*(1-exp(x))))))

a(n) = n!*sum(k=1, n\4, k^(k-1)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

a(n) = n! * Sum_{k=1..floor(n/4)} k^(k-1) * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A350746 Triangle read by rows: T(n,k) is the number of labeled quasi-loop-threshold graphs on vertex set [n] with k components, for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

2, 3, 4, 16, 18, 8, 133, 155, 72, 16, 1521, 1810, 910, 240, 32, 22184, 26797, 14145, 4180, 720, 64, 393681, 480879, 262514, 83230, 16520, 2016, 128, 8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256

Offset: 1

David Galvin, Jan 13 2022

The family of quasi-loop-threshold graphs is the smallest family of looped graphs that contains K_1 (a single vertex) and K^loop_1 (a single looped vertex), and is closed under taking unions and adding looped dominating vertices (looped, and adjacent to everything previously added).

			Triangle begins:
        2;
        3,        4;
       16,       18,       8;
      133,      155,      72,      16;
     1521,     1810,     910,     240,     32;
    22184,    26797,   14145,    4180,    720,    64;
   393681,   480879,  262514,   83230,  16520,  2016,  128;
  8233803, 10144283, 5675866, 1888873, 409360, 58912, 5376, 256;
  ...

D. Galvin, G. Wesley and B. Zacovic, Enumerating threshold graphs and some related graph classes, arXiv:2110.08953 [math.CO], 2021.

Row sums are A038052.

Except at n=1, the first column is A048802 (A048802 takes value 1 at n=1).

qltconn[0] = 0; qltconn[1] = 2; qltconn[n_] := qltconn[n] = Sum[StirlingS2[n, k]*(k^(k - 1)), {k, 1, n}] (*qltconn is the number of connected quasi loop threshold graphs on n vertices*); T[n_, l_] := T[n, l] := (Factorial[n]/Factorial[l])*Coefficient[(Sum[(qltconn[k]*(x^k))/Factorial[k], {k, 1, n}])^l, x, n]; Table[T[n, l], {n, 1, 12}, {l, 1, n}]

See Section 1.4 of Galvin, Wesley and Zacovic link for two methods to compute T(n,k).

cp's OEIS Frontend

A357346 E.g.f. satisfies A(x) = (exp(x * exp(A(x))) - 1) * exp(A(x)).

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A355179 Expansion of e.g.f. -LambertW(x^2 * (1 - exp(x)))/2.

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A355180 Expansion of e.g.f. -LambertW(x^3 * (1 - exp(x)))/6.

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A356000 Expansion of e.g.f. -LambertW((1 - exp(2*x))/2).

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A356001 Expansion of e.g.f. -LambertW((1 - exp(3*x))/3).

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A216858 Number of connected functions from {1,2,...,n} into a subset of {1,2,...,n} summed over all subsets.

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Mathematica

PARI

Formula

A355181 Expansion of e.g.f. -LambertW(x^2/2 * (1 - exp(x))).

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PARI

PARI

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A355308 Expansion of e.g.f. -LambertW(x^3/6 * (1 - exp(x))).

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