A349525 -id:A349525 - 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

A349524 a(n) = Sum_{k=0..n} (2k+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 6, 65, 1059, 23232, 642859, 21507733, 844701160, 38108248719, 1942394699283, 110401966739110, 6923805346540685, 474957822716470901, 35377953843680999326, 2843665890900123673997, 245340865605247369255751, 22614510471168438300336440, 2217985444621941684970200607

Offset: 0

Vaclav Kotesovec, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..348

Cf. A000110, A008277, A052880, A349504, A349525.

b:= proc(n, m) option remember; `if`(n=0,
     (2*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[(2*k+1)^(k-1)*StirlingS2[n, k], {k, 0, n}], {n, 0, 20}]
nmax = 20; CoefficientList[Series[Sqrt[-LambertW[2*(-E^x + 1)]/(2*(E^x - 1))], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = sum(k=0, n, (2*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, (2*k+1)^(k-1)*x^k/prod(j=1, k, 1-j*x))) \\ Seiichi Manyama, Nov 20 2021

E.g.f.: sqrt(-LambertW(2*(-exp(x) + 1)) / (2*(exp(x) - 1))).
E.g.f.: exp(-LambertW(2 - 2*exp(x))/2).
a(n) ~ c * d^n * n! / n^(3/2), where d = 1/log(1 + 1/(2*exp(1))) and c = sqrt(exp(1) * (1 + 2*exp(1)) * log(1 + 1/(2*exp(1))) / (2*Pi))/2 = 0.3428481589262346912499652905097648170872882109000404115070292580887155335...
a(n) ~ sqrt(1 + 2*exp(1)) * n^(n-1) / (2*exp(n - 1/2) * log(1 + 1/(2*exp(1)))^(n - 1/2)).
E.g.f. satisfies: log(A(x)) = (exp(x) - 1) * A(x)^2.
G.f.: Sum_{k>=0} (2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x). - Seiichi Manyama, Nov 20 2021

A349505 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^3).

Original entry on oeis.org

1, 1, 6, 81, 1668, 46740, 1659102, 71386602, 3611376360, 210083758704, 13817649943440, 1013979735381888, 82134894774767832, 7279520816638839600, 700730732176038359208, 72803537907677356262760, 8120227815636769492383168

Offset: 0

Seiichi Manyama, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..336

Cf. A008275, A033917, A264408, A349504, A349525.

nmax = 20; CoefficientList[Series[(-LambertW[-3*Log[1 + x]]/(3*Log[1 + x]))^(1/3), {x, 0, nmax}], x]*Range[0, nmax]! (* Vaclav Kotesovec, Nov 20 2021 *)

a(n) = sum(k=0, n, (3*k+1)^(k-1)*stirling(n, k, 1));

N=20; x='x+O('x^N); Vec(serlaplace(sum(k=0, N, (3*k+1)^(k-1)*log(1+x)^k/k!)))

a(n) = Sum_{k=0..n} (3*k+1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = Sum_{k>=0} (3*k+1)^(k-1) * (log(1+x))^k / k!.
From Vaclav Kotesovec, Nov 20 2021: (Start)
E.g.f.: (-LambertW(-3*log(1 + x)) / (3*log(1 + x)))^(1/3).
a(n) ~ n^(n-1) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2) * exp(n + exp(-1)/6 - 5/6)). (End)

A349683 E.g.f. satisfies: log(A(x)) = exp(x*A(x)^3) - 1.

Original entry on oeis.org

1, 1, 8, 131, 3303, 113137, 4909829, 258275887, 15974450676, 1136164798581, 91366516437475, 8197719659916195, 811910298234609913, 87984131560596043801, 10355660409438349522396, 1315550191540192189444535, 179413108433279983993509731

Offset: 0

Seiichi Manyama, Nov 25 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..328

Cf. A349525, A349598.

nterms=20;Table[Sum[(3n+1)^(k-1)*StirlingS2[n,k],{k,0,n}],{n,0,nterms-1}] (* Paolo Xausa, Nov 25 2021 *)

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

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

a(n) ~ s * n^(n-1) / (3 * sqrt(1 + r*s^3) * exp(n) * r^n), where r = LambertW(1/3)/exp(1/LambertW(1/3) - 3) = 0.106691814639676411952403096776061... and s = exp(1/(3*LambertW(1/3)) - 1) = 1.341591995635184131204677967393502... are roots of the system of equations exp(r*s^3) = 1 + log(s), 3*r*s^3*exp(r*s^3) = 1. - Vaclav Kotesovec, Nov 26 2021

A349528 a(n) = Sum_{k=0..n} (-1)^(n-k) * (3k+1)^(k-1) Stirling2(n,k).

Original entry on oeis.org

1, 1, 6, 80, 1645, 45962, 1627080, 69817575, 3522349232, 204343964292, 13403304111515, 980876342339456, 79235384391436316, 7003257362607771709, 672285536392973397658, 69656231091367157111844, 7747832754070176901631621

Offset: 0

Seiichi Manyama, Nov 20 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..336
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A008277, A058864, A196556, A349525, A349527.

a[n_] := Sum[(-1)^(n - k)*(3*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Nov 21 2021 *)

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

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

my(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)))

E.g.f. satisfies: log(A(x)) = (1 - exp(-x)) * A(x)^3.
E.g.f.: exp( -LambertW(3 * (exp(-x) - 1))/3 ).
G.f.: Sum_{k>=0} (3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ sqrt(3*exp(1) - 1) * sqrt(log(3/(3-exp(-1)))) * n^(n-1) / (3 * exp(n - 1/3) * (1 + log(3/(3*exp(1) - 1)))^n). - Vaclav Kotesovec, Nov 21 2021

A357336 E.g.f. satisfies A(x) = (exp(x) - 1) * exp(3 * A(x)).

Original entry on oeis.org

0, 1, 7, 100, 2257, 70021, 2768740, 133164109, 7546722487, 492531820066, 36381833190223, 3000677194970137, 273342303933512362, 27256107730344331879, 2952882035628632383975, 345384835617231362018764, 43378466647737203462409829, 5822506028894124326533926193

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, A357335.

Cf. A349525, A356001.

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

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

E.g.f.: -LambertW(3 * (1 - exp(x)))/3.
a(n) = Sum_{k=1..n} (3 * k)^(k-1) * Stirling2(n,k).
a(n) ~ sqrt(1 + 3*exp(1)) * n^(n-1) / (3 * exp(n) * log(1 + exp(-1)/3)^(n - 1/2)). - Vaclav Kotesovec, Nov 14 2022
E.g.f.: Series_Reversion( log(1 + x * exp(-3*x)) ). - Seiichi Manyama, Sep 09 2024

A367201 E.g.f. satisfies log(A(x)) = (exp(xA(x)) - 1) A(x)^3.

Original entry on oeis.org

1, 1, 10, 200, 6167, 258607, 13748744, 886397829, 67211684890, 5861684458896, 578088714806497, 63617223837958309, 7728596914020856162, 1027393177458209939977, 148344954037140113652010, 23119776330887635387231580, 3868359765874829925197165527

Offset: 0

Seiichi Manyama, Nov 10 2023

nonn

Cf. A349525, A367163.
Cf. A349557, A355762.
Cf. A367199, A367200.

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

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

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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A349524 a(n) = Sum_{k=0..n} (2*k+1)^(k-1) * Stirling2(n,k).

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A349505 E.g.f. satisfies: A(x) = (1 + x)^(A(x)^3).

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A349683 E.g.f. satisfies: log(A(x)) = exp(x*A(x)^3) - 1.

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A349528 a(n) = Sum_{k=0..n} (-1)^(n-k) * (3*k+1)^(k-1) * Stirling2(n,k).

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A357336 E.g.f. satisfies A(x) = (exp(x) - 1) * exp(3 * A(x)).

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A367201 E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * A(x)^3.

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

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A349524 a(n) = Sum_{k=0..n} (2k+1)^(k-1) Stirling2(n,k).

A349528 a(n) = Sum_{k=0..n} (-1)^(n-k) * (3k+1)^(k-1) Stirling2(n,k).

A367201 E.g.f. satisfies log(A(x)) = (exp(xA(x)) - 1) A(x)^3.

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