A349524 -id:A349524 - cp's OEIS frontend

A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

1, 1, 0, 2, -9, 72, -710, 8563, -121814, 1997502, -37097739, 769687954, -17644355410, 442894514285, -12081668234012, 355889274553166, -11258683640579857, 380701046875217492, -13702507978018209458, 523049811008797507683, -21105565578064063658754

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

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

Cf. A120980, A349561, A349585.

Cf. A008277, A052880, A349524, A349525.

b:= proc(n, m) option remember; `if`(n=0,
     (1-m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022

a[n_] := Sum[If[k == 1, 1, (-k + 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* Amiram Eldar, Nov 23 2021 *)

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x)))

a(n) = Sum_{k=0..n} (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(exp(x) - 1) ).
G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
a(n) ~ -(-1)^n * sqrt(exp(1) - 1) * n^(n-1) / (exp(n+1) * (1 - log(exp(1) - 1))^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021

A349525 a(n) = Sum_{k=0..n} (3k+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 8, 122, 2847, 90112, 3611162, 175352515, 10009442658, 656934750150, 48744407335597, 4035143806865514, 368706775967717518, 36861117438297883213, 4002400525694764367212, 469049713401827161071110, 59010099414303871987517111, 7932542361585921797125908876

Offset: 0

Vaclav Kotesovec, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..331

Cf. A000110, A008277, A052880, A349505, A349524.

b:= proc(n, m) option remember; `if`(n=0,
     (3*m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 29 2022

Table[Sum[(3*k+1)^(k-1)*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[(-LambertW[3*(-E^x + 1)]/(3*(E^x - 1)))^(1/3), {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = sum(k=0, n, (3*k+1)^(k-1)*stirling(n, k, 2)); \\ Seiichi Manyama, Nov 20 2021

N=20; x='x+O('x^N); Vec(sum(k=0, N, (3*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Nov 20 2021

E.g.f.: (-LambertW(3*(-exp(x) + 1)) / (3*(exp(x) - 1)))^(1/3).
E.g.f.: exp(-LambertW(3 - 3*exp(x))/3).
a(n) ~ c * d^n * n! / n^(3/2), where d = 1/log(1 + 1/(3*exp(1))) and c = exp(1/3) * sqrt((1 + 3*exp(1)) * log(1 + 1/(3*exp(1))) / (2*Pi))/3 = 0.190981550465823640438134269765128596177617920807463710992027181154754728...
a(n) ~ sqrt(1 + 3*exp(1)) * n^(n-1) / (3*exp(n - 1/3) * log(1 + 1/(3*exp(1)))^(n - 1/2)).
E.g.f. satisfies: log(A(x)) = (exp(x) - 1) * A(x)^3.
G.f.: Sum_{k>=0} (3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x). - Seiichi Manyama, Nov 20 2021

A349598 E.g.f. satisfies: log(A(x)) = exp(x*A(x)^2) - 1.

Original entry on oeis.org

1, 1, 6, 71, 1279, 31142, 958127, 35674921, 1560207964, 78410153193, 4453247964775, 282086867840252, 19718661737739301, 1507855981764016549, 125211854842018500134, 11220898483255456505555, 1079389691811367897870339, 110936313685240067472613726

Offset: 0

Seiichi Manyama, Nov 22 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..343

Cf. A001761, A001763, A349599.

Cf. A349524, A349601.

a[n_] := Sum[(2*n + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 23 2021 *)

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

a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * Stirling2(n,k).
a(n) ~ s * n^(n-1) / (2 * sqrt(1 + r*s^2) * exp(n) * r^n), where r = 0.1513832219344136560178112221696108323993292386502... and s = 1.52429184135463908701026733917578550814344591549... are roots of the system of equations (1 + log(s))*2*r*s^2 = 1, 2*r*s^2*exp(r*s^2) = 1. - Vaclav Kotesovec, Nov 25 2021
Equivalently, a(n) ~ n^(n-1) / (2*sqrt(1 + LambertW(1/2)) * LambertW(1/2)^n * exp(3*n + 1 - (n + 1/2)/LambertW(1/2))). - Vaclav Kotesovec, Nov 26 2021

A349504 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).

Original entry on oeis.org

1, 1, 4, 36, 484, 8840, 203868, 5691308, 186612592, 7031373264, 299397454080, 14218443479328, 745142534904480, 42717896158340832, 2659373970144454080, 178666030775042040000, 12884568940594969258752, 992750028716940749121792

Offset: 0

Seiichi Manyama, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..356

Cf. A008275, A033917, A264407, A349505, A349524.

nmax = 20; CoefficientList[Series[Sqrt[-LambertW[-2*Log[1 + x]]/(2*Log[1 + x])], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 20 2021 *)

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

N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (2*k+1)^(k-1)*log(1+x)^k/k!)))

a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = Sum_{k>=0} (2*k+1)^(k-1) * (log(1+x))^k / k!.
From Vaclav Kotesovec, Nov 20 2021: (Start)
E.g.f.: sqrt(-LambertW(-2*log(1 + x)) / (2*log(1 + x))).
a(n) ~ n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2) * exp(n + exp(-1)/4 - 1)). (End)

A349527 a(n) = Sum_{k=0..n} (-1)^(n-k) * (2k+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 4, 35, 469, 8502, 194807, 5402497, 175985390, 6587650497, 278674144201, 13148017697608, 684554867667117, 38988819551585477, 2411411875573335044, 160951864352781351959, 11531509389384310870257

Offset: 0

Seiichi Manyama, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..356
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A008277, A058864, A196555, A349524, A349528.

a[n_] := Sum[(-1)^(n - k) * (2*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Nov 27 2021 *)

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

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

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (2*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

E.g.f. satisfies: log(A(x)) = (1 - exp(-x)) * A(x)^2.
E.g.f.: exp( -LambertW(2 * (exp(-x) - 1))/2 ).
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ sqrt(2*exp(1) - 1) * sqrt(log(2/(2-exp(-1)))) * n^(n-1) / (2 * exp(n - 1/2) * (1 + log(2/(2*exp(1) - 1)))^n). - Vaclav Kotesovec, Nov 21 2021

A357335 E.g.f. satisfies A(x) = (exp(x) - 1) * exp(2 * A(x)).

Original entry on oeis.org

0, 1, 5, 49, 757, 16081, 435477, 14345297, 556857973, 24894290257, 1259621627349, 71165987957329, 4440821632449077, 303338709537825105, 22512353926895739797, 1803812930088064925265, 155195078834104237961717, 14270228623788585753803089

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, A357336.
Cf. A349524, A356000.
Cf. A357347.

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

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

E.g.f.: -LambertW(2 * (1 - exp(x)))/2.
a(n) = Sum_{k=1..n} (2 * k)^(k-1) * Stirling2(n,k).
a(n) ~ sqrt(1 + 2*exp(1)) * n^(n-1) / (2 * exp(n) * log(1 + exp(-1)/2)^(n - 1/2)). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( log(1 + x * exp(-2*x)) ). - Seiichi Manyama, Sep 09 2024

A356199 a(n) = Sum_{k=0..n} (nk+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 6, 122, 5991, 556152, 84245291, 18956006323, 5940695613628, 2474958812797662, 1323229303771318595, 883245295259143164922, 719968321620942410875645, 703829776430361739799683993, 812798413118207226439408790038, 1094718407894086754989907938078190

Offset: 0

Alois P. Heinz, Jul 29 2022

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..215

Cf. A000110, A008277, A048993, A052880, A349524, A349525, A349583, A349654, A349655.

b:= proc(n, k, m) option remember; `if`(n=0,
     (k*m+1)^(m-1), m*b(n-1, k, m)+b(n-1, k, m+1))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..19);

b[n_, k_, m_] := b[n, k, m] = If[n == 0,
   (k*m+1)^(m-1), m*b[n-1, k, m] + b[n-1, k, m+1]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 14 2023, after Alois P. Heinz *)

a(n) = sum(k=0, n, (n*k+1)^(k-1) * stirling(n, k, 2)); \\ Michel Marcus, Aug 04 2022

a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).
a(n) = [x^n] Sum_{k>=0} (n*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
a(n) = n! * [x^n] 1/exp(LambertW((1 - exp(x))*n)/n) for n > 0, a(0) = 1.
a(n) ~ exp(exp(-1)/2) * n^(2*n - 2). - Vaclav Kotesovec, Aug 07 2022

cp's OEIS Frontend

A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

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A349525 a(n) = Sum_{k=0..n} (3*k+1)^(k-1) * Stirling2(n,k).

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A349598 E.g.f. satisfies: log(A(x)) = exp(x*A(x)^2) - 1.

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A349504 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^2).

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A349527 a(n) = Sum_{k=0..n} (-1)^(n-k) * (2*k+1)^(k-1) * Stirling2(n,k).

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A357335 E.g.f. satisfies A(x) = (exp(x) - 1) * exp(2 * A(x)).

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A356199 a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).

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A349525 a(n) = Sum_{k=0..n} (3k+1)^(k-1) Stirling2(n,k).

A349527 a(n) = Sum_{k=0..n} (-1)^(n-k) * (2k+1)^(k-1) Stirling2(n,k).

A356199 a(n) = Sum_{k=0..n} (nk+1)^(k-1) Stirling2(n,k).