A349505 -id:A349505 - cp's OEIS frontend

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

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

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)

A264408 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^3).

Original entry on oeis.org

1, 1, 8, 123, 2880, 91380, 3670422, 178637046, 10220188584, 672297233904, 49998246211440, 4148400045354048, 379922492043352728, 38069325641907527760, 4143052833221549219112, 486645149764644576491160, 61364015736691449905270208, 8267876533375590035644494336, 1185420892076469924434688605760

Offset: 0

Paul D. Hanna, Nov 18 2015

nonn

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 123*x^3/3! + 2880*x^4/4! + 91380*x^5/5! + 3670422*x^6/6! + 178637046*x^7/7! + 10220188584*x^8/8! +...
where A(x) = 1/(1-x)^(A(x)^3).
From a LambertW identity,
A(x) = 1 - log(1-x) + 7*log(1-x)^2/2! - 10^2*log(1-x)^3/3! + 13^3*log(1-x)^4/4! - 16^4*log(1-x)^5/5! + 19^5*log(1-x)^6/6! +...
Also,
A(x) = 1 + x*A(x)^3 + x^2*A(x)^3*(A(x)^3+1)/2! + x^3*A(x)^3*(A(x)^3+1)*(A(x)^3+2)/3! + x^4*A(x)^3*(A(x)^3+1)*(A(x)^3+2)*(A(x)^3+3)/4! +...

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

Cf. A052813, A264407, A349505.

Table[Sum[Abs[StirlingS1[n, k]] * (3*k+1)^(k-1), {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Nov 18 2015 *)

{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A^3 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

a(n) = sum(k=0, n, abs(stirling(n, k, 1))*(3*k+1)^(k-1));
for(n=0, 20, print1(a(n), ", "))

E.g.f. also satisfies:
(1) A(x) = ( LambertW(3*log(1-x)) / (3*log(1-x)) )^(1/3).
(2) A(x) = Sum_{n>=0} (3*n+1)^(n-1) * (-log(1-x))^n / n!.
(3) A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x)^3 + k).
(4) A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)| * A(x)^(3*k).
a(n) = Sum_{k=0..n} |Stirling1(n, k)| * (3*k+1)^(k-1).
a(n) ~ n^(n-1) / (sqrt(3) * exp(n - n*exp(-1)/3 - 5/6) * (exp(exp(-1)/3) - 1)^(n-1/2)). - Vaclav Kotesovec, Nov 18 2015

A357338 E.g.f. satisfies A(x) = log(1 + x) * exp(3 * A(x)).

Original entry on oeis.org

0, 1, 5, 65, 1302, 35904, 1260372, 53796168, 2704942440, 156602951568, 10260496538640, 750563024381928, 60636437884772208, 5362045857366832152, 515154874732515894744, 53432840588453561773080, 5950904875941534263739648, 708296073287989866587094528

Offset: 0

Seiichi Manyama, Sep 24 2022

nonn

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

Cf. A277489, A357337.

Cf. A349505.

nmax = 20; A[_] = 0;
Do[A[x_] = Log[1 + x]*Exp[3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)

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

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

E.g.f.: -LambertW(-3 * log(1 + x))/3.
a(n) = Sum_{k=1..n} (3 * k)^(k-1) * Stirling1(n,k).
a(n) ~ n^(n-1) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2) * exp(n - 1/2 + exp(-1)/6)). - Vaclav Kotesovec, Oct 04 2022
E.g.f.: Series_Reversion( exp(x * exp(-3*x)) - 1 ). - Seiichi Manyama, Sep 10 2024

A362794 E.g.f. satisfies A(x) = (1+x)^(A(x)^x).

Original entry on oeis.org

1, 1, 0, 6, 0, 170, -120, 12446, -35336, 1832400, -12172320, 469680552, -5524990416, 189586178184, -3321122831208, 111608536026360, -2599887499382400, 90253048158627072, -2595580675897337856, 95720854442948910720, -3237436187047116892800

Offset: 0

Seiichi Manyama, May 04 2023

sign

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

Cf. A033917, A362795.
Cf. A362796, A362799.
Cf. A349504, A349505.

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

E.g.f.: exp( -LambertW(-x * log(1+x)) / x ) = (1+x)^exp( -LambertW(-x * log(1+x)) ).

E.g.f.: Sum_{k>=0} (k*x + 1)^(k-1) * (log(1+x))^k / k!.

A367199 E.g.f. satisfies A(x) = (1 + x*A(x))^(A(x)^3).

Original entry on oeis.org

1, 1, 8, 141, 3852, 143460, 6780642, 388851960, 26235133992, 2036243259648, 178742696099040, 17509589369568648, 1893647907646728120, 224106838102512869400, 28809018473999642686584, 3997516614926297143604760, 595518793080901690966354368

Offset: 0

Seiichi Manyama, Nov 10 2023

nonn

Cf. A349505, A367157.
Cf. A162655, A162656.
Cf. A367200, A367201.

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

a(n) = Sum_{k=0..n} (n+3*k+1)^(k-1) * Stirling1(n,k).

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A264408 E.g.f. satisfies: A(x) = 1/(1-x)^(A(x)^3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A357338 E.g.f. satisfies A(x) = log(1 + x) * exp(3 * A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A362794 E.g.f. satisfies A(x) = (1+x)^(A(x)^x).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A367199 E.g.f. satisfies A(x) = (1 + x*A(x))^(A(x)^3).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

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