A349583 -id:A349583 - cp's OEIS frontend

A349654 E.g.f. satisfies: A(x)^2 * log(A(x)) = exp(x) - 1.

1, 1, -2, 17, -213, 3712, -82773, 2250565, -72218912, 2671680015, -111950278213, 5240764049094, -271082407059027, 15353947287972373, -945097225235334538, 62820021683240176445, -4484426869618973019249, 342169496779859317566456

Offset: 0

Seiichi Manyama, Nov 23 2021

sign

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

Cf. A349650, A349652, A349656.

Cf. A008277, A349583, A349655.

b:= proc(n, m) option remember; `if`(n=0,
     (1-2*m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 29 2022

a[n_] := Sum[(-2*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (-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)))

a(n) = Sum_{k=0..n} (-2*k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = 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) ~ -(-1)^n * sqrt(2*exp(1) - 1) * sqrt(log(2) - log(2 - exp(-1))) * n^(n-1) / (2 * exp(n + 1/2) * (log(2) - log(2*exp(1) - 1) + 1)^n). - Vaclav Kotesovec, Nov 24 2021

A120980 E.g.f. satisfies: A(x)^A(x) = 1 + x.

Original entry on oeis.org

1, 1, -2, 9, -68, 740, -10554, 185906, -3891320, 94259952, -2592071760, 79748398752, -2713685928744, 101184283477680, -4102325527316184, 179674073609647080, -8454031849605513024, 425281651659459346944, -22777115050468598701248

Offset: 0

Paul D. Hanna, Jul 20 2006

sign

G. C. Greubel, Table of n, a(n) for n = 0..375

Cf. A008275, A052813, A052807, A349561, A349583, A349585.

CoefficientList[Series[Log[1+x]/LambertW[Log[1+x]], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
Table[StirlingS1[n, 0] + StirlingS1[n, 1] + Sum[(-1)^(k + 1)*StirlingS1[n, k]*(k - 1)^(k - 1), {k, 2, n}], {n,0,50}] (* G. C. Greubel, Jun 21 2017 *)
CoefficientList[Series[Exp[LambertW[Log[1+x]]], {x, 0, 25}], x]* Range[0, 25]! (* G. C. Greubel, Jun 22 2017 *)

{a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0); A[ #A]=-Vec(Ser(A)^Ser(A))[ #A]);n!*A[n+1]}

x='x+O('x^50); Vec(serlaplace(exp(lambertw(log(1+x))))) \\ G. C. Greubel, Jun 22 2017

E.g.f.: A(x) = log(1+x)/LambertW(log(1+x)).
log(A(x)) = LambertW(log(1+x)).
E.g.f.: A(x) = 1/G(-x) where G(x) = g.f. of A052813.
E.g.f. of A052807 = -log(A(-x)) = -log(1-x)/A(-x).
a(n) = Sum_{k=0..n} (-1)^(k+1)*Stirling1(n,k)*(k-1)^(k-1). - Vladeta Jovovic, Jul 22 2006
|a(n)| ~ exp((exp(-1)-1)*n+3/2) * n^(n-1) / (exp(exp(-1))-1)^(n-1/2). - Vaclav Kotesovec, Jul 09 2013

A349601 E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)^2) - 1.

Original entry on oeis.org

1, 1, 4, 32, 391, 6462, 134974, 3412187, 101323674, 3457536144, 133333945461, 5734792007584, 272197255745078, 14133109419794601, 796883164532719216, 48489515568651113516, 3167153388603620859695, 221021628292403019655418

Offset: 0

Seiichi Manyama, Nov 22 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..361

Cf. A216135, A349600, A349602.

Cf. A349583, A349588.

Table[Sum[(2*n - k + 1)^(k-1) * StirlingS2[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 25 2021 *)

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

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

a(n) ~ s * n^(n-1) / (sqrt(2*(1 + 2*r*s^2) - 2/(1 + log(s))) * exp(n) * r^n), where r = 0.2229533052706631261980294005821031136702825459439... and s = 1.759796045489338472919926226485178994146849909897... are roots of the system of equations exp(r*s^2) = 1 + s*log(s), 2*exp(r*s^2)*r*s = 1 + log(s). - Vaclav Kotesovec, Nov 25 2021

A349655 E.g.f. satisfies: A(x)^3 * log(A(x)) = exp(x) - 1.

Original entry on oeis.org

1, 1, -4, 50, -981, 26632, -924526, 39114343, -1952373450, 112321286934, -7318049389727, 532602419776770, -42825957593127770, 3770431528821292441, -360734325565272740984, 37267364164988863692782, -4134667018838875759388749, 490302545213321842575157140

Offset: 0

Seiichi Manyama, Nov 23 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..337
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349651, A349653, A349657.

Cf. A008277, A349583, A349654.

b:= proc(n, m) option remember; `if`(n=0,
     (1-3*m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 29 2022

a[n_] := Sum[(-3*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 27 2021 *)

a(n) = sum(k=0, n, (-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)))

a(n) = Sum_{k=0..n} (-3*k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = 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) ~ -(-1)^n * sqrt(3*exp(1) - 1) * sqrt(log(3) - log(3 - exp(-1))) * n^(n-1) / (3 * exp(n + 1/3) * (log(3) - log(3*exp(1) - 1) + 1)^n). - Vaclav Kotesovec, Nov 24 2021

A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).

Original entry on oeis.org

1, 1, -2, 8, -59, 642, -9112, 158839, -3279880, 78250188, -2117569181, 64082989720, -2144319848772, 78609355884893, -3133061858717806, 134884905211588892, -6238095343894356675, 308427209934965151158, -16234730389499986865092, 906409067599064528054343

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..378
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A120980, A349561, A349583.

Cf. A008277, A058864, A349527, A349528.

b:= proc(n, m) option remember; `if`(n=0,
     (m-1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> (-1)^(n-1)*b(n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 03 2022

a[n_] := (-1)^(n - 1) * Sum[If[k == 1, 1, (k - 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 23 2021 *)

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

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

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

A356902 E.g.f. satisfies A(x) * log(A(x)) = x * (exp(x) - 1).

Original entry on oeis.org

1, 0, 2, 3, -8, -55, 276, 4417, -13488, -639567, -248300, 141842921, 797525400, -43103642855, -584650622724, 16366430341185, 436555007091616, -6909610676492959, -368240758971238620, 2371795171252419385, 354876368637537736680, 1050192150132691993161

Offset: 0

Seiichi Manyama, Sep 03 2022

sign

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

Cf. A355843, A356797, A356798, A356904.

Cf. A349583.

nmax = 21; A[_] = 1;
Do[A[x_] = Exp[(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\2, (-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))

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

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

a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(x * (exp(x) - 1)) ).
E.g.f.: A(x) = -x * (1 - exp(x))/LambertW(-x * (1 - exp(x))).

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

A357011 E.g.f. satisfies A(x) * log(A(x)) = exp(x * A(x)^3) - 1.

Original entry on oeis.org

1, 1, 6, 74, 1407, 36357, 1190476, 47254783, 2205546706, 118378505742, 7184030384361, 486440226752911, 36358328607088010, 2973464028723984551, 264119772408892921774, 25321946948812001539166, 2606224408648404660237647, 286624141573198517220290837

Offset: 0

Seiichi Manyama, Sep 08 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..340

Cf. A349583, A349588, A349601.

Cf. A216136, A356960, A356973.

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

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

A357244 E.g.f. satisfies A(x) * log(A(x)) = 2 * (exp(x) - 1).

Original entry on oeis.org

1, 2, -2, 22, -266, 4614, -102442, 2777030, -88914730, 3283693254, -137408080298, 6425417730758, -332055079469610, 18792899306652358, -1156017201432075946, 76796076655220486854, -5479395288838822143786, 417905042599836811225798, -33928512587303405767179178

Offset: 0

Seiichi Manyama, Sep 19 2022

sign

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

Cf. A349583, A357245.

Cf. A356908.

nmax = 18; A[_] = 1;
Do[A[x_] = Exp[(2*(Exp[x] - 1))/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

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

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

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

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

a(n) = Sum_{k=0..n} 2^k * (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (2 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(2 * (exp(x) - 1)) ).
E.g.f.: A(x) = 2 * (exp(x) - 1)/LambertW(2 * (exp(x) - 1)).

A357245 E.g.f. satisfies A(x) * log(A(x)) = 3 * (exp(x) - 1).

Original entry on oeis.org

1, 3, -6, 84, -1599, 42906, -1477716, 62171661, -3090518556, 177237143040, -11518529575857, 836601742598628, -67156626492464064, 5904119985344031639, -564188922815428792914, 58225175660113940932032, -6453955474121138652732903, 764716767229825444834522086

Offset: 0

Seiichi Manyama, Sep 19 2022

sign

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

Cf. A349583, A357244.

nmax = 17; A[_] = 1;
Do[A[x_] = Exp[(3*(Exp[x] - 1))/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

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

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

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

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

a(n) = Sum_{k=0..n} 3^k * (-k+1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (3 * (exp(x) - 1))^k / k!.
E.g.f.: A(x) = exp( LambertW(3 * (exp(x) - 1)) ).
E.g.f.: A(x) = 3 * (exp(x) - 1)/LambertW(3 * (exp(x) - 1)).

cp's OEIS Frontend

A349654 E.g.f. satisfies: A(x)^2 * log(A(x)) = exp(x) - 1.

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A120980 E.g.f. satisfies: A(x)^A(x) = 1 + x.

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A349601 E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)^2) - 1.

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A349655 E.g.f. satisfies: A(x)^3 * log(A(x)) = exp(x) - 1.

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A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).

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A356902 E.g.f. satisfies A(x) * log(A(x)) = x * (exp(x) - 1).

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

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