A216136 -id:A216136 - cp's OEIS frontend

A141209 E.g.f. satisfies A(x)^A(x) = 1/(1 - x*A(x)).

1, 1, 2, 9, 64, 620, 7626, 113792, 1997192, 40316544, 920271840, 23438308872, 658947505272, 20270099889624, 677226678369528, 24420959694718680, 945370712175873216, 39103903755819561984, 1721215383181421110848, 80329148928437231089152

Offset: 0

Paul D. Hanna, Jul 01 2008

nonn

Generally, if e.g.f. A(x) satisfies A(x)^A(x) = 1/(1-x*A(x)^p), then a(n) ~ s*sqrt((s^s-1)/(p*(s^s-1)*(p*s^s-1)-s)) * n^(n-1) * (s^(p+s)/(s^s-1))^n / exp(n), where s is the root of the equation (1+log(s))*s = (s^s-1)*p. - Vaclav Kotesovec, Dec 28 2013

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 620*x^5/5! +...

G. C. Greubel, Table of n, a(n) for n = 0..385

Cf. A216135, A216136, A229237, A008275 (Stirling1), A141209 (A162655), A191908.

Table[Sum[(n-k+1)^(k-1)*Abs[StirlingS1[n,k]], {k, 0, n}], {n, 0, 20}]  (* Vaclav Kotesovec, Sep 17 2013 *)
E^((2*r-1)/(1-r))*(1+(1-r)*E^(r/(r-1)))/.FindRoot[E^(r/(1-r))==(r-1)/r*(r+LambertW[-1,-r*E^(-r)]), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n)/n, Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=local(A=1+x);for(i=0,n,A=exp(-log(1-x*(A+O(x^n)))/A));n!*polcoeff(A,n)}

{a(n,m=1)=sum(k=0,n,m*(n-k+m)^(k-1)*polcoeff(prod(j=1,n-1,1+j*x),n-k))} \\ Paul D. Hanna, Jul 08 2009

{a(n,m=1)=n!*sum(k=0,n,m*(n-k+m)^(k-1)*polcoeff((-log(1-x+x*O(x^n))/x)^k/k!,n-k))} \\ Paul D. Hanna, Jul 08 2009

{a(n,m=1)=sum(k=0,n,m*(n-k+m)^(k-1)*(-1)^(n-k)*stirling(n,k,1))} \\ Paul D. Hanna, Jul 08 2009

From Paul D. Hanna, Jul 08 2009: (Start)
(1) a(n) = Sum_{k=0..n} (n-k+1)^(k-1) *(-1)^(n-k) *Stirling1(n,k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
(2) a(n,m) = Sum_{k=0..n} m*(n-k+m)^(k-1) *(-1)^(n-k) *Stirling1(n,k) ;
which is equivalent to the following:
(3) a(n,m) = Sum_{k=0..n} m*(n-k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1+j*x)};
(4) a(n,m) = n!*Sum_{k=0..n} m*(n-k+m)^(k-1) * {[x^(n-k)] (-log(1-x)/x)^k/k!}.
(End)
Limit_{n->oo} a(n)^(1/n)/n = exp((2*r-1)/(1-r))*(1+(1-r)*exp(r/(r-1))) = 0.97848198198076..., where r = 0.42324001455512542... is the root of the equation exp(r/(1-r)) = (r-1)/r*(r + LambertW(-1,-r*exp(-r))). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ s*sqrt((s^s-1)/((s^s-1)^2-s)) * n^(n-1) * (s^(1+s)/(s^s-1))^n / exp(n), where s = 2.083029805648017585241865819... is the root of the equation (1+log(s))*s = (s^s-1). - Vaclav Kotesovec, Dec 28 2013

A349561 E.g.f. satisfies: A(x)^A(x) = 1/(1 - x).

Original entry on oeis.org

1, 1, 0, 3, -8, 100, -834, 11438, -159928, 2762352, -52322160, 1124320032, -26509832040, 686751503568, -19306448087640, 586539826169880, -19131996548499264, 667157522614934016, -24762890955027112128, 974824890777753840576, -40566428716555791936000

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

			A(x) - 1 = x + 3*x^3/6 - 8*x^4/24 + ... = x + x^3/2 - x^4/3 + ... .
A(x)^A(x) = (1 + (A(x) - 1))^(1 + (A(x) - 1)) = Sum_{k>=0} A005727(k) * (A(x) - 1)^k / k! = 1 + 1 * (x + x^3/2 - x^4/3 + ... )/1! + 2 * (x + x^3/2 - x^4/3 + ... )^2/2! + 3 * (x + x^3/2 - x^4/3 + ... )^3/3! + ...  = 1 + x + x^2 + x^3 + ... = 1/(1 - x).

Seiichi Manyama, Table of n, a(n) for n = 0..398
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A005727, A120980, A141209, A216135, A216136, A229237, A305819.

Join[{1}, Table[(n-1)! - (-1)^n*Sum[(k-1)^(k-1)*StirlingS1[n, k], {k, 2, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 22 2021 *)

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

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

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

a(n) = (-1)^(n-1) * Sum_{k=0..n} (k-1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = -Sum_{k>=0} (k-1)^(k-1) * (log(1-x))^k / k!.
E.g.f.: A(x) = -log(1-x)/LambertW(-log(1-x)).
a(n) ~ -(-1)^n * n^(n-1) / ((exp(exp(-1)) - 1)^(n - 1/2) * exp(n + exp(-1)/2 + 1/2)). - Vaclav Kotesovec, Nov 22 2021

A216135 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^2).

Original entry on oeis.org

1, 1, 4, 33, 424, 7440, 165846, 4487966, 142930376, 5237697744, 217106129040, 10043789510832, 513016686849624, 28676264198255856, 1741205465305623240, 114124985340571809480, 8030944551164700156096, 603905270121593669417472, 48328182913534662635924544

Offset: 0

Paul D. Hanna, Sep 01 2012

nonn

More generally, if G(x) = 1/(1 - x*G(x)^p)^(G(x)^q), then
[x^n/n! ] G(x)^m = Sum_{k=0..n} m*(p*n+q*k+m)^(k-1) * (-1)^(n-k)*Stirling1(n,k), and
[x^n/n! ] log(G(x)) = Sum_{k=1..n} (p*n+q*k)^(k-1) * (-1)^(n-k)*Stirling1(n,k).
Generally, if e.g.f. A(x) satisfies A(x)^A(x) = 1/(1-x*A(x)^p), then a(n) ~ s*sqrt((s^s-1)/(p*(s^s-1)*(p*s^s-1)-s)) * n^(n-1) * (s^(p+s)/(s^s-1))^n / exp(n), where s is the root of the equation (1+log(s))*s = (s^s-1)*p. - Vaclav Kotesovec, Dec 28 2013

			E.g.f. A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 424*x^4/4! + 7440*x^5/5! +...
where
A(x)^A(x) = 1 + x + 6*x^2/2! + 60*x^3/3! + 864*x^4/4! + 16360*x^5/5! +...
1/(1-x*A(x)^2) = 1 + x + 6*x^2/2! + 60*x^3/3! + 864*x^4/4! + 16360*x^5/5! +...

Seiichi Manyama, Table of n, a(n) for n = 0..357

Cf. A141209, A216136, A229237.

Table[Sum[(2*n-k+1)^(k-1)*(-1)^(n-k)*StirlingS1[n,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 17 2013 *)

a(n, m=1)=sum(k=0, n, m*(2*n-k+m)^(k-1)*(-1)^(n-k)*stirling(n, k, 1));
for(n=0,21,print1(a(n),", "))

{a(n, m=1)=sum(k=0, n, m*(2*n-k+m)^(k-1)*polcoeff(prod(j=1, n-1, 1+j*x), n-k))}
for(n=0,21,print1(a(n),", "))

{a(n)=local(A=1+x); for(i=0, n, A=exp(-log(1-x*(A^2+x*O(x^n)))/A)); n!*polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

(1) a(n) = Sum_{k=0..n} (2*n-k+1)^(k-1)* (-1)^(n-k)* Stirling1(n,k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
(2) a(n,m) = Sum_{k=0..n} m*(2*n-k+m)^(k-1) * (-1)^(n-k) *Stirling1(n,k) ;
which is equivalent to the following:
(3) a(n,m) = Sum_{k=0..n} m*(2*n-k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1+j*x)};
(4) a(n,m) = n!*Sum_{k=0..n} m*(2*n-k+m)^(k-1) * {[x^(n-k)] (-log(1-x)/x)^k/k!}.
Limit n->infinity a(n)^(1/n)/n = exp(2*(1-r)/(r-2))*(2-r+exp(r/(2-r))) = 1.7802115440907..., where r = 0.655269699533064... is the root of the equation exp(r/(2-r)) = ((r-2)/r)*(r + LambertW(-1,-r*exp(-r))). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ s*sqrt((s^s-1)/(2*(s^s-1)*(2*s^s-1)-s)) * n^(n-1) * (s^(2+s)/(s^s-1))^n / exp(n), where s = 1.627893875694537903318580987... is the root of the equation (1+log(s))*s = 2*(s^s-1). - Vaclav Kotesovec, Dec 28 2013

A229237 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^4).

Original entry on oeis.org

1, 1, 8, 135, 3544, 126980, 5778606, 319234454, 20755549256, 1552791269232, 131408062049040, 12411898074678432, 1294418444771718168, 147733436055601473168, 18315901821846419101416, 2451257290708213030681080, 352217918432527724627871936, 54082428426583359310449351168

Offset: 0

Vaclav Kotesovec, Sep 17 2013

nonn

Generally, for A(x)^A(x) = 1/(1-x*A(x)^p) is limit n->infinity a(n)^(1/n)/n = exp(p*(1-r)/(r-p))*(p-r+exp(r/(p-r))), where r is the root of the equation exp(r/(p-r)) = (r-p)/r*(r + LambertW(-1,-r*exp(-r))

Generally, if e.g.f. A(x) satisfies A(x)^A(x) = 1/(1-x*A(x)^p), then a(n) ~ s*sqrt((s^s-1)/(p*(s^s-1)*(p*s^s-1)-s)) * n^(n-1) * (s^(p+s)/(s^s-1))^n / exp(n), where s is the root of the equation (1+log(s))*s = (s^s-1)*p. Compared with my previous result, limit n->infinity a(n)^(1/n)/n = s^(p+s)/(s^s-1)/exp(1). - Vaclav Kotesovec, Dec 28 2013

Seiichi Manyama, Table of n, a(n) for n = 0..326

Cf. A141209, A216135, A216136, A349561.

Table[Sum[(4*n-k+1)^(k-1)*(-1)^(n-k)*StirlingS1[n,k],{k,0,n}],{n,0,20}]
p=4; E^(p*(1-r)/(r-p))*(p-r+E^(r/(p-r)))/.FindRoot[E^(r/(p-r))==(r-p)/r*(r+LambertW[-1,-r*E^(-r)]), {r,1/2}, WorkingPrecision->50] (* program for numerical value of the limit n->infinity a(n)^(1/n)/n *)

Limit n->infinity a(n)^(1/n)/n = exp(4*(1-r)/(r-4))*(4-r+exp(r/(4-r))) = 3.635561077783029..., where r = 0.8373821681637... is the root of the equation exp(r/(4-r)) = (r-4)/r*(r + LambertW(-1,-r*exp(-r))

a(n) ~ s*sqrt((s^s-1)/(4*(s^s-1)*(4*s^s-1)-s)) * n^(n-1) * (s^(4+s)/(s^s-1))^n / exp(n), where s = 1.3031377498774256189193761312... is the root of the equation (1+log(s))*s = 4*(s^s-1). - Vaclav Kotesovec, Dec 28 2013

A356973 E.g.f. satisfies log(A(x)) = (exp(x * A(x)^3) - 1) * A(x).

Original entry on oeis.org

1, 1, 10, 206, 6555, 283777, 15577332, 1036984027, 81191314678, 7311591070938, 744577308572189, 84608911909469235, 10613728203840498210, 1456899252646375490851, 217215453964895439271178, 34956361099228031471844962, 6039398076840098381458042875

Offset: 0

Seiichi Manyama, Sep 07 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..319

Cf. A052880, A349557, A356972.

Cf. A216136, A356960.

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).

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).

cp's OEIS Frontend

A141209 E.g.f. satisfies A(x)^A(x) = 1/(1 - x*A(x)).

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A349561 E.g.f. satisfies: A(x)^A(x) = 1/(1 - x).

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A216135 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^2).

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A229237 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^4).

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A356973 E.g.f. satisfies log(A(x)) = (exp(x * A(x)^3) - 1) * A(x).

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

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