A141209 -id:A141209 - cp's OEIS frontend

A349588 E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)) - 1.

1, 1, 2, 8, 47, 367, 3592, 42317, 583522, 9223872, 164482761, 3267077365, 71540314562, 1712334954865, 44479256704898, 1246241906483516, 37465750470667023, 1202986323660907447, 41089436549405467096, 1487622596267089224901, 56907111260864275384346

Offset: 0

Seiichi Manyama, Nov 22 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..403

Cf. A141209, A349587, A349589.

Cf. A349557.

a[n_] := Sum[(n - 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, (n-k+1)^(k-1)*stirling(n, k, 2));

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

a(n) ~ s * n^(n-1) / (sqrt(r*s - 1/(1 + log(s))) * exp(n) * r^n), where r = 0.4858893246242883887847088396703818017675758048583... and s = 3.016426175038226058288579473351450432292607021364... are roots of the system of equations exp(r*s) = 1 + s*log(s), exp(r*s)*r = 1 + log(s). - Vaclav Kotesovec, Nov 25 2021

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

A162655 E.g.f. satisfies: A(x) = (1 + x*A(x))^A(x).

Original entry on oeis.org

1, 1, 4, 33, 416, 7100, 153234, 4004000, 122919208, 4336955424, 172946624880, 7692618593352, 377615317473624, 20278301717340888, 1182581903027279832, 74428445506232769240, 5028336618916834615104, 362962785521720282899200

Offset: 0

Paul D. Hanna, Jul 08 2009

nonn

Contribution from Paul D. Hanna, Jul 19 2009: (Start)
More generally, if G(x) = (1 + x*G(x)^p)^(G(x)^q), then
[x^n/n! ] G(x)^m = Sum_{k=0..n} m*(pn+qk+m)^(k-1) * Stirling1(n,k), and
[x^n/n! ] log(G(x)) = Sum_{k=1..n} (pn+qk)^(k-1) * Stirling1(n,k). (End)

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 416*x^4/4! + 7100*x^5/5! +...
log(A(x)) = A(x)*log(1 + x*A(x)) where
log(A(x)) = x + 3*x^2/2! + 23*x^3/3! + 278*x^4/4! + 4624*x^5/5! + 98064*x^6/6! +...
log(1 + x*A(x)) = x + x^2/2! + 8*x^3/3! + 90*x^4/4! + 1444*x^5/5! + 29880*x^6/6! +...

Seiichi Manyama, Table of n, a(n) for n = 0..359

Cf. A008275 (Stirling1), A141209 (variant).

Cf. A162863. [From Paul D. Hanna, Jul 19 2009]

Table[Sum[(n+k+1)^(k-1) * StirlingS1[n,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)

{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))}

{a(n,m=1)=sum(k=0,n,m*(n+k+m)^(k-1)*n!/k!*polcoeff((log(1+x+x*O(x^n))/x)^k,n-k))}

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

(1) a(n) = Sum_{k=0..n} (n+k+1)^(k-1) * 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) * 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!}.
a(n) ~ s^2*sqrt(r*(1+r*s)/(1+r*s*(1+s)*(2+r*s))) * n^(n-1) / (exp(n)*r^n), where r = 0.21551711955114319212... and s = 1.7128732151580576508... are roots of the system of equations s*(r*s/(1+r*s) + log(1+r*s)) = 1, (1+r*s)^s = s. - Vaclav Kotesovec, Jul 15 2014

A216136 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^3).

Original entry on oeis.org

1, 1, 6, 75, 1456, 38560, 1296786, 52935680, 2542934024, 140557741488, 8787984793440, 613224873661752, 47245653830341176, 3983499665690137944, 364844394810538703256, 36070922050704987248280, 3828821598701561543783616, 434302348322255060713797120

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, 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))). - Vaclav Kotesovec, Sep 17 2013
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

			E.g.f. A(x) = 1 + x + 6*x^2/2! + 75*x^3/3! + 1456*x^4/4! + 38560*x^5/5! +...
where
A(x)^A(x) = 1 + x + 8*x^2/2! + 114*x^3/3! + 2388*x^4/4! + 66480*x^5/5! +...
1/(1-x*A(x)^3) = 1 + x + 8*x^2/2! + 114*x^3/3! + 2388*x^4/4! + 66480*x^5/5! +...

Seiichi Manyama, Table of n, a(n) for n = 0..338

Cf. A141209, A216135, A229237.

Table[Sum[(3*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*(3*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*(3*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^3+x*O(x^n)))/A)); n!*polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

(1) a(n) = Sum_{k=0..n} (3*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*(3*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*(3*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*(3*n-k+m)^(k-1) * {[x^(n-k)] (-log(1-x)/x)^k/k!}.
Limit n->infinity a(n)^(1/n)/n = exp(3*(1-r)/(r-3))*(3-r+exp(r/(3-r))) = 2.685525290558..., where r = 0.77397865498224... is the root of the equation exp(r/(3-r)) = ((r-3)/r)*(r + LambertW(-1,-r*exp(-r))). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ s*sqrt((s^s-1)/(3*(s^s-1)*(3*s^s-1)-s)) * n^(n-1) * (s^(3+s)/(s^s-1))^n / exp(n), where s = 1.4158017407588097722625060603... is the root of the equation (1+log(s))*s = 3*(s^s-1). - Vaclav Kotesovec, Dec 28 2013

A162656 E.g.f. satisfies: A(x) = (1 + x*A(x))^(A(x)^2).

Original entry on oeis.org

1, 1, 6, 78, 1564, 42660, 1475028, 61838000, 3048628336, 172850367744, 11081419939680, 792685606699008, 62593631986243488, 5408185369666834560, 507526235156395322112, 51407696054804232576000

Offset: 0

Paul D. Hanna, Jul 08 2009

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 78*x^3/3! + 1564*x^4/4! + 42660*x^5/5! +...
log(A(x)) = A(x)^2*log(1 + x*A(x)) where
log(A(x)) = x + 5*x^2/2! + 62*x^3/3! + 1210*x^4/4! + 32464*x^5/5! +...
log(1 + x*A(x)) = x + x^2/2! + 14*x^3/3! + 246*x^4/4! + 6284*x^5/5! +...

Seiichi Manyama, Table of n, a(n) for n = 0..337

Cf. A008275 (Stirling1), variants: A162655, A141209.

Table[Sum[(n+2*k+1)^(k-1) * StirlingS1[n,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)

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

{a(n,m=1)=sum(k=0,n,m*(n+2*k+m)^(k-1)*n!/k!*polcoeff((log(1+x+x*O(x^n))/x)^k,n-k))}

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

(1) a(n) = Sum_{k=0..n} (n+2k+1)^(k-1) * 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+2k+m)^(k-1) * Stirling1(n,k) ;
which is equivalent to the following:
(3) a(n,m) = Sum_{k=0..n} m*(n+2k+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+2k+m)^(k-1) * {[x^(n-k)] (log(1+x)/x)^k/k!}.
a(n) ~ s^2*sqrt(r*s*(1+r*s) / (2+r*s*(4+3*s^2+2*r*s*(1+s^2)))) * n^(n-1) / (exp(n)*r^n), where r = 0.1337990328596883051... and s = 1.408676495644569372... are roots of the system of equations s^2*(r*s + 2*(1+r*s)*log(1+r*s)) = 1+r*s, (1+r*s)^(s^2) = s. - Vaclav Kotesovec, Jul 15 2014

A349587 E.g.f. satisfies: A(x)^A(x) = 1 + x*A(x).

Original entry on oeis.org

1, 1, 0, -3, 4, 60, -294, -2800, 34504, 197568, -6087360, -9146808, 1488986808, -5886157992, -469973309064, 5492298353880, 177826238321856, -4277426240130048, -72353540601814464, 3537861051231290880, 22847222673714931200, -3226666120379253611136

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..421

Cf. A141209, A349588, A349589.

Cf. A120980, A162655, A162656.

a[n_] := Sum[(n - k + 1)^(k - 1)*StirlingS1[n, k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Nov 23 2021 *)

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

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

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

A349589 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x*A(x)).

Original entry on oeis.org

1, 1, 0, -4, -3, 87, 230, -4583, -27216, 434928, 4871719, -62913079, -1240374960, 12230778601, 426135019232, -2759957884648, -189393687667107, 479371576805751, 105233549909615798, 233116575802412969, -71022416772836562008, -574100485456271792020

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425

Cf. A141209, A349587, A349588.

Cf. A349558.

a[n_] := Sum[(-1)^(n - k)*(n - k + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, Nov 23 2021 *)

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

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

A162863 E.g.f. satisfies: A(x) = (1 + x*A(x)^2)^A(x).

Original entry on oeis.org

1, 1, 6, 75, 1448, 38020, 1265454, 51069326, 2423671144, 132284727792, 8164129781280, 562204918658592, 42737232766827576, 3554783958154270608, 321149971312286643240, 31316069883727673961240

Offset: 0

Paul D. Hanna, Jul 15 2009

nonn

More generally, if G(x) = (1 + x*G(x)^p)^(G(x)^q), then
[x^n/n! ] G(x)^m = Sum_{k=0..n} m*(pn+qk+m)^(k-1) * Stirling1(n,k), and
[x^n/n! ] log(G(x)) = Sum_{k=1..n} (pn+qk)^(k-1) * Stirling1(n,k).

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 75*x^3/3! + 1448*x^4/4! +...
A(x)^2 = 1 + 2*x + 14*x^2/2! + 186*x^3/3! + 3712*x^4/4! +...
log(A(x)) = A(x)*log(1 + x*A(x)^2) where
log(A(x)) = x + 5*x^2/2! + 59*x^3/3! + 1106*x^4/4! + 28524*x^5/5! +...
log(1 + x*A(x)^2) = x + 3*x^2/2! + 32*x^3/3! + 570*x^4/4! + 14264*x^5/5! +...

G. C. Greubel, Table of n, a(n) for n = 0..338

Cf. A008275 (Stirling1), variants: A162655, A141209.

[(&+[(2*n+k+1)^(k-1)*StirlingFirst(n,k) : k in [0..n]]): n in [0..20]]; // G. C. Greubel, Oct 24 2018

Table[Sum[(2*n+k+1)^(k-1) * StirlingS1[n,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)

{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))}

{a(n,m=1)=n!*sum(k=0,n,m*(2*n+k+m)^(k-1)*polcoeff((log(1+x+x*O(x^n))/x)^k/k!,n-k))}

{Stirling1(n,k)=n!*polcoeff(binomial(x,n),k)}
{a(n,m=1)=sum(k=0,n,m*(2*n+k+m)^(k-1)*Stirling1(n,k))}

(1) a(n) = Sum_{k=0..n} (2n+k+1)^(k-1) * 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*(2n+k+m)^(k-1) * Stirling1(n,k) ;
(3) a(n,m) = Sum_{k=0..n} m*(2n+k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1-j*x)} ;
(4) a(n,m) = Sum_{k=0..n} m*(2n+k+m)^(k-1) * n!*{[x^(n-k)] (log(1+x)/x)^k/k!}.
Let log(A(x)) = Sum_{n>=0} L(n)*x^n/n!, then
(5) L(n) = Sum_{k=1..n} (2n+k)^(k-1) * Stirling1(n,k).
a(n) ~ s^2*sqrt(r*s*(1+r*s^2)/(1+r*s^2*(2+s*(6+r*s*(1+2*s))))) * n^(n-1) / (exp(n)*r^n), where r = 0.1389785143116673015... and s = 1.435128235324409145... are roots of the system of equations s*(2*r*s^2/(1+r*s^2) + log(1+r*s^2)) = 1, (1+r*s^2)^s = s. - Vaclav Kotesovec, Jul 15 2014

cp's OEIS Frontend

A349588 E.g.f. satisfies: A(x) * log(A(x)) = exp(x*A(x)) - 1.

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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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A162655 E.g.f. satisfies: A(x) = (1 + x*A(x))^A(x).

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A216136 E.g.f. A(x) satisfies: A(x)^A(x) = 1/(1 - x*A(x)^3).

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A162656 E.g.f. satisfies: A(x) = (1 + x*A(x))^(A(x)^2).

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A349587 E.g.f. satisfies: A(x)^A(x) = 1 + x*A(x).