A177380 -id:A177380 - cp's OEIS frontend

A238274 Decimal expansion of abs(LambertW(-1)).

1, 3, 7, 4, 5, 5, 7, 0, 1, 0, 7, 4, 3, 7, 0, 7, 4, 8, 6, 5, 3, 0, 0, 9, 3, 0, 5, 6, 7, 6, 9, 6, 6, 2, 6, 7, 2, 3, 4, 4, 2, 9, 7, 6, 3, 6, 5, 3, 7, 6, 2, 6, 5, 0, 0, 1, 0, 9, 6, 5, 7, 1, 0, 6, 3, 2, 4, 2, 1, 6, 6, 9, 5, 6, 5, 6, 4, 8, 7, 1, 5, 1, 7, 1, 3, 8, 3, 6, 7, 0, 0, 6, 4, 1, 9, 6, 4, 9, 4, 0, 0, 6, 8, 2, 4

Offset: 1

Vaclav Kotesovec, Feb 24 2014

			1.37455701074370748653...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Stanislav Sykora, Fixed points of the mappings exp(z) and -exp(z) in C, 2016.

Cf. A009306, A030178, A177380, A185221.

RealDigits[N[Abs[LambertW[-1]], 105]][[1]]

A185221 E.g.f. is solution to y = 1 + log(1 + x*y) in powers of x.

Original entry on oeis.org

1, 1, 1, -1, -10, -6, 294, 1350, -14624, -197568, 703800, 34790040, 100585968, -7259053296, -85604489712, 1588693382640, 46549054391040, -216669088277760, -24865626969568512, -159153249738896640, 13379663931502199040

Offset: 0

Michael Somos, Jan 24 2012

sign

			y = 1 + x + 1/2*x^2 - 1/6*x^3 - 5/12*x^4 - 1/20*x^5 + 49/120*x^6 + 15/56*x^7 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A177380, A238274.

Cf. A097718, A138013.

a(n):=if n=0 then 1 else sum(binomial(n+k+1,n) * sum((-1)^(j) * binomial(k+1,j) * sum((-1)^i * i! * binomial(j+i-1,j-1) * stirling1(n,i), i,1,n), j,1,k+1), k,0,n) / (n+1); /* Vladimir Kruchinin, Mar 29 2013 */

{a(n) = local(A); if( n<0, 0, A = 1 + O(x); for( k=1, n, A = 1 + log(1 + x * A)); n! * polcoeff( A, n))}

E.g.f. is solution to y = y' * (1 - x + x*y).
a(n) = sum(k=0..n, binomial(n+k+1,n) * sum(j=1..k+1, (-1)^(j) * binomial(k+1,j) * sum(i=1..n, (-1)^i * i! * binomial(j+i-1,j-1) * stirling1(n,i)))) / (n+1), n>0, a(0)=1. [Vladimir Kruchinin, Mar 29 2013]
Lim sup n->infinity (|a(n)|/n!)^(1/n) = abs(LambertW(-1)) = 1.37455701074370748653... (see A238274). - Vaclav Kotesovec, Feb 24 2014
a(n) = n! * Sum_{k=0..n} Stirling1(n,k)/(n-k+1)!. - Seiichi Manyama, Nov 07 2023

A247075 Expansion of e.g.f.: x^2G'(x)/G(x)^2, where G(x) satisfies G(x) = x(1+log(1+G(x))).

Original entry on oeis.org

1, 0, -1, -2, 12, 96, -220, -7440, -15624, 813120, 7340112, -104165280, -2442773520, 8815815360, 855578733984, 4653629425536, -317564443445760, -5591544140206080, 110965435244017920, 4730495445765296640, -16883238483957574656

Offset: 0

Vladimir Kruchinin, Nov 17 2014

sign

G. C. Greubel, Table of n, a(n) for n = 0..425

Cf. A177380.

[(&+[Factorial(j)*Binomial(n-1,j)*StirlingFirst(n,j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Mar 08 2023

A:= n -> add(k!*binomial(n-1,k)*combinat:-stirling1(n,k),k=0..n):
seq(A(n),n=0..30); # Robert Israel, Nov 17 2014

Table[Sum[StirlingS1[n, k] k! Binomial[n-1, k], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Nov 17 2014 *)

a(n):=sum(k!*binomial(n-1,k)*stirling1(n,k),k,0,n);

def A247075(n): return sum( (-1)^(n-k)*factorial(k)*binomial(n-1,k)*stirling_number1(n,k) for k in range(n+1))
[A247075(n) for n in range(21)] # G. C. Greubel, Mar 08 2023

a(n) = Sum_{k=0..n} k!*binomial(n-1,k)*Stirling1(n,k).

E.g.f.: x^2*G'(x)/G(x)^2 where G(x) = Series_Reversion(x/(1 + log(1+x))); see A177380. - Paul D. Hanna, Nov 17 2014

A177379 E.g.f. satisfies: A(x) = 1/(1-x - x*log(A(x))).

Original entry on oeis.org

1, 1, 4, 27, 260, 3270, 50904, 946134, 20462896, 505137312, 14020517520, 432340670520, 14667108820704, 542979374426736, 21784934875431168, 941691211940974320, 43634507604383543040, 2157698329617806488320

Offset: 0

Paul D. Hanna, May 15 2010

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 260*x^4/4! +...
Log(A(x)) = G(x) - 1 where G(x) = e.g.f. of A138013 begins:
G(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! + 1694*x^5/5! + ...
and satisfies: exp(1 - G(x)) = 1 - x*G(x).
Contribution from _Paul D. Hanna_, Jul 16 2010: (Start)
Given e.g.f. A(x), and A179424 = Riordan array (A(x),x*A(x)) where the g.f. of column k in A179424 equals A(x)^(k+1):
1;
1, 1;
4/2!, 2, 1;
27/3!, 10/2!, 3, 1;
260/4!, 78/3!, 18/2!, 4, 1;
3270/5!, 832/4!, 159/3!, 28/2!, 5, 1;
...
then the matrix log of A179424 equals the triangular matrix:
0;
1, 0;
1, 2, 0;
1, 2, 3, 0;
1, 2, 3, 4, 0;
1, 2, 3, 4, 5, 0; ...
in which column k equals k+1 in row n for n>k>=0.
(End)

Cf. A177380, A138013, A226572.

Cf. A179424. [From Paul D. Hanna, Jul 16 2010]

CoefficientList[1/(1-InverseSeries[Series[x/(1-Log[1-x]),{x,0,20}],x]),x]*Range[0,20]! (* Vaclav Kotesovec, Jan 11 2014 *)

{a(n)=n!*polcoeff(1/(1-serreverse(x/(1-log(1-x+x*O(x^n))))),n)}

/* Using matrix log of Riordan array (A(x),x*A(x)): */
{a(n)=local(L=matrix(n+1,n+1,r,c,if(r>c,c)),M=sum(m=0,#L,L^m/m!));n!*M[n+1,1]} \\ Paul D. Hanna, Jul 16 2010

/* From A(x) = (1 + x*A'(x)/A(x))*(1 - x*A(x))/(1-x): */
{a(n)=local(A=1+x);for(k=2,n,A=A-polcoeff((1+x*deriv(A)/A)*(1-x*A)/(1-x+x*O(x^n)),k)*x^k/(k-1));n!*polcoeff(A,n)} \\ Paul D. Hanna, Jul 16 2010

E.g.f.: A(x) = 1/(1 - Series_Reversion(x/(1 - log(1-x)))).
...
Let G(x) = e.g.f. of A138013, then:
. A(x) = exp(G(x) - 1),
. A(x) = 1/(1 - x*G(x))
where G(x) = 1 - log(1 - x*G(x)).
...
Let F(x) = e.g.f. of A177380, then:
. [x^n] A(x)^(-n+1)/(-n+1) = A177380(n)/n! for n>1,
. [x^n] F(x)^(n+1)/(n+1) = a(n)/n! for n>=0,
. A(x) = F(x*A(x)) and A(x/F(x)) = F(x),
. A(x) = (1/x)*Series_Reversion(x/F(x))
where F(x) = 1+x + x*log(F(x)).
Contribution from Paul D. Hanna, Jul 16 2010: (Start)
E.g.f. satisfies: A(x) = (1 + x*A'(x)/A(x))*(1 - x*A(x))/(1-x).
...
Let A_n(x) denote the n-th iteration of x*A(x) with G = x/(1-x), then:
. A(x) = 1 + G + G*Dx(G)/2! + G*Dx(G*Dx(G))/3! + G*Dx(G*Dx(G*Dx(G)))/4! +...
. A_n(x)/x = 1 + n*G + n^2*G*Dx(G)/2! + n^3*G*Dx(G*Dx(G))/3! + n^4*G*Dx(G*Dx(G*Dx(G)))/4! +...
where Dx(F) = d/dx(x*F).
...
Given e.g.f. A(x), the matrix log of the Riordan array (A(x),x*A(x)) equals the matrix L defined by L(n,k)=k+1 and L(n,n)=0, for n>=0, n>k.
(End)
a(n) ~ sqrt(s-1) * n^(n-1) * s^(n+1) / exp(n), where s = -LambertW(-1,-exp(-2)) = 3.14619322062... (see A226572). - Vaclav Kotesovec, Jan 11 2014

A308565 a(n) = Sum_{k=0..n} binomial(n,k) * Stirling1(n,k) * k!.

Original entry on oeis.org

1, 1, 0, -6, -12, 140, 1020, -5208, -117264, -2448, 17756640, 117905040, -3177424800, -56997933408, 523176632160, 25824592321920, 31907065317120, -12118922683971840, -151839867298498560, 5619086944920958464, 172859973799199892480, -1989399401447725854720, -170925579909303883614720

Offset: 0

Ilya Gutkovskiy, Jun 07 2019

sign

Cf. A000312, A000522, A006252, A177380, A247075, A277759, A317274.

Table[Sum[Binomial[n, k] StirlingS1[n, k] k!, {k, 0, n}], {n, 0, 22}]
Table[n! SeriesCoefficient[(1 + Log[1 + x])^n, {x, 0, n}], {n, 0, 22}]

a(n) = n! * [x^n] (1 + log(1 + x))^n.

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A238274 Decimal expansion of abs(LambertW(-1)).

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A185221 E.g.f. is solution to y = 1 + log(1 + x*y) in powers of x.

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A247075 Expansion of e.g.f.: x^2*G'(x)/G(x)^2, where G(x) satisfies G(x) = x*(1+log(1+G(x))).

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A177379 E.g.f. satisfies: A(x) = 1/(1-x - x*log(A(x))).

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A308565 a(n) = Sum_{k=0..n} binomial(n,k) * Stirling1(n,k) * k!.

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A247075 Expansion of e.g.f.: x^2G'(x)/G(x)^2, where G(x) satisfies G(x) = x(1+log(1+G(x))).