A366233 -id:A366233 - cp's OEIS frontend

A161633 E.g.f. satisfies A(x) = 1/(1 - xexp(xA(x))).

1, 1, 4, 27, 268, 3525, 57966, 1146061, 26500552, 702069129, 20974309210, 697754762001, 25584428686620, 1025230366195789, 44579963354153878, 2090676600895922565, 105191995364927688976, 5652501986238910061073, 323083811850594613809714, 19573120681427758058921881

Offset: 0

Paul D. Hanna, Jun 18 2009

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3525*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...

Seiichi Manyama, Table of n, a(n) for n = 0..374

Cf. A006153, A161630 (e.g.f. = exp(x*A(x))), A213644, A364980, A364981.

Cf. A366232, A366233, A366234, A366235.

Flatten[{1,Table[n!*Sum[Binomial[n+1,k]/(n+1) * k^(n-k)/(n-k)!,{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)

a(n,m=1)=n!*sum(k=0,n,binomial(n+m,k)*m/(n+m)*k^(n-k)/(n-k)!)

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x)*exp(x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).
(3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.
a(n) = n! * Sum_{k=0..n} binomial(n+1,k)/(n+1) * k^(n-k)/(n-k)!.
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n! * Sum_{k=0..n} binomial(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.
a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - Vaclav Kotesovec, Jan 10 2014

A212917 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^3) ).

Original entry on oeis.org

1, 1, 3, 31, 469, 9681, 254701, 8131999, 305626329, 13218345793, 646712664121, 35315446759671, 2129341219106773, 140506900034640049, 10071589943109973461, 779311468200041101711, 64742128053980794659121, 5747587082198264156035329, 543023929087191507383612785

Offset: 0

Paul D. Hanna, May 30 2012

nonn

From Vaclav Kotesovec, Jul 15 2014: (Start)
Generally, if e.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^p)), p>=1, then
r = 4*LambertW(sqrt(p)/2)^2 / (p*(1+2*LambertW(sqrt(p)/2))),
A(r) = (sqrt(p)/(2*LambertW(sqrt(p)/2)))^(2/p),
a(n) ~ p^(n-1+1/p) * (1+2*LambertW(sqrt(p)/2))^(n+1/2) * n^(n-1) / (sqrt(1+LambertW(sqrt(p)/2)) * exp(n) * 2^(2*n+2/p) * LambertW(sqrt(p)/2)^(2*n+2/p-1/2)).
(End)

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + ...
such that, by definition:
log(A(x))/x = 1 + x*A(x)^3 + x^2*A(x)^6 + x^3*A(x)^9 + x^4*A(x)^12 + ...
Related expansions:
log(A(x)) = x/(1-x*A(x)^3) = x + 2*x^2/2! + 24*x^3/3! + 348*x^4/4! + 7140*x^5/5! + 186750*x^6/6! + ... + n*A366233(n-1)*x^n/n! + ...
A(x)^3 = 1 + 3*x + 15*x^2/2! + 153*x^3/3! + 2421*x^4/4! + 51363*x^5/5! + 1375029*x^6/6! + ...
A(x)^6 = 1 + 6*x + 48*x^2/2! + 576*x^3/3! + 9864*x^4/4! + 221256*x^5/5! + 6156756*x^6/6! + ...

G. C. Greubel, Table of n, a(n) for n = 0..349
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265

Cf. A161630, A212722, A245265.

Cf. A366233 (log).

Table[Sum[n! * (1 + 3*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *)

{a(n, m=1)=if(n==0, 1, sum(k=0, n, n!/k!*m*(m+3*(n-k))^(k-1)*binomial(n-1, n-k)))}

{a(n, m=1)=local(A=1+x); for(i=1, n, A=exp(x/(1-x*A^3+x*O(x^n)))); n!*polcoeff(A^m, n)}
for(n=0, 21, print1(a(n), ", "))

a(n) = Sum_{k=0..n} n! * (1 + 3*(n-k))^(k-1)/k! * C(n-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} n! * m*(m + 3*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ 3^(n-2/3) * n^(n-1) * (1+2*c)^(n+1/2) / (sqrt(1+c) * 2^(2*n+2/3) * exp(n) * c^(2*n+1/6)), where c = LambertW(sqrt(3)/2) = 0.5166154518588324282494... . - Vaclav Kotesovec, Jul 15 2014

A366232 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(2xA(x)).

Original entry on oeis.org

1, 1, 6, 54, 728, 13000, 290352, 7800016, 245115264, 8826560640, 358463525120, 16212238054144, 808215885708288, 44035925223746560, 2603618739995621376, 166031767704180111360, 11359670347331723952128, 830065763154102656204800, 64518486557995327748898816

Offset: 0

Paul D. Hanna, Oct 05 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum_{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
(2) F(x) = (2/x) * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
(3) F(x) = (3/x) * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
(4) F(x) = (4/x) * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
(5) F(x) = (k/x) * Sum_{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 54*x^3/3! + 728*x^4/4! + 13000*x^5/5! + 290352*x^6/6! + 7800016*x^7/7! + 245115264*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(2*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+1*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(-0*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(-1*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-2*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-3*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-4*x*A(x))/6! + ...
and
A(x) = 1 + 3*1*3^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 3*2*4^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 3*3*5^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 3*4*6^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 3*5*7^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ... + A212722(n)*x^n/n! + ...

Cf. A161633, A366233, A366234, A366235.

Cf. A365772 (dual), A212722 (exp(x*A(x))).

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(2*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)

/* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 2^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(2*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

a(n) = n! * Sum_{k=0..n} binomial(n+1, n-k)/(n+1) * 2^k * (n-k)^k / k!.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 2^k * (n-k)^k/k!.
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x) * exp(2*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(2*x)) ).
(3) A( x/(1 + x*exp(2*x)) ) = 1 + x*exp(2*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-2)*x*A(x)) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
(4.b) A(x) = 1 + 2 * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
(4.c) A(x) = 1 + 3 * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
(4.d) A(x) = 1 + 4 * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-(n+1)*x*A(x)).
a(n) ~ n^(n-1) * (1 + 2*LambertW(1/sqrt(2)))^(n + 3/2) / (sqrt(1 + LambertW(1/sqrt(2))) * 2^(n+2) * exp(n) * LambertW(1/sqrt(2))^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(4xA(x)).

Original entry on oeis.org

1, 1, 10, 126, 2392, 60600, 1916304, 72917488, 3246171520, 165609099648, 9529240349440, 610657739172096, 43136025287678976, 3330356645773880320, 279024535906794539008, 25214258236430338160640, 2444656672390982922502144, 253144081975231633923342336

Offset: 0

Paul D. Hanna, Oct 05 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum_{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
(2) F(x) = (2/x) * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
(3) F(x) = (3/x) * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
(4) F(x) = (4/x) * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
(5) F(x) = (k/x) * Sum_{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 126*x^3/3! + 2392*x^4/4! + 60600*x^5/5! + 1916304*x^6/6! + 72917488*x^7/7! + 3246171520*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(4*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+3*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+2*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(+1*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-0*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-1*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-2*x*A(x))/6! + ...
and
A(x) = 1 + 5*1*5^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 5*2*6^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 5*3*7^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 5*4*8^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 5*5*9^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ... + A245265(n)*x^n/n! + ...

Cf. A161633, A366232, A366233, A366235.

Cf. A365774 (dual), A245265 (exp(x*A(x))).

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(4*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)

/* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 4^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(4*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

a(n) = n! * Sum_{k=0..n} binomial(n+1, n-k)/(n+1) * 4^k * (n-k)^k / k!.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 4^k * (n-k)^k/k!.
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x) * exp(4*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(4*x)) ).
(3) A( x/(1 + x*exp(4*x)) ) = 1 + x*exp(4*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-4)*x*A(x)) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-4)*x*A(x)).
(4.b) A(x) = 1 + 2 * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).
(4.c) A(x) = 1 + 3 * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
(4.d) A(x) = 1 + 4 * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
(4.e) A(x) = 1 + 5 * Sum_{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
(4.f) A(x) = 1 + 6 * Sum_{n>=1} n*(n+5)^(n-2) * x^n/n! * A(x)^n * exp(-(n+1)*x*A(x)).
a(n) ~ (1 + 2*LambertW(1))^(n + 3/2) * n^(n-1) / (4 * sqrt(1 + LambertW(1)) * exp(n) * LambertW(1)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A365773 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 3x*A(x))^2.

Original entry on oeis.org

1, 1, 7, 46, 325, 2446, 19234, 156115, 1298077, 11000584, 94668508, 825087418, 7267943962, 64602794647, 578726742481, 5219620390558, 47357456920165, 431941341136552, 3958215409319608, 36425213089790932, 336475535026075180, 3118885520601252016, 29000562051786329512

Offset: 0

Paul D. Hanna, Oct 04 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

			G.f.: A(x) = 1 + x + 7*x^2 + 46*x^3 + 325*x^4 + 2446*x^5 + 19234*x^6 + 156115*x^7 + 1298077*x^8 + 11000584*x^9 + 94668508*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2
also
A(x) = 1 + 1^0*x^1*A(x)^1/(1 + (-2)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-1)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + 0*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 1*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 2*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 3*x*A(x))^7 + ...
and
A(x) = 1 + 4*1*4^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 4*2*5^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 4*3*6^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 4*4*7^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 4*5*8^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A365770, A109081, A365772, A365774, A365775.

Cf. A366233 (dual).

{a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 3^k)}
for(n=0,30, print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 3*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 3^k.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 3^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 3*x)^2) ).
(3) A( x/(1 + x/(1 - 3*x)^2) ) = 1 + x/(1 - 3*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-3)*x*A(x))^(n+1) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n * A(x)^n / (1 + (n+1)*x*A(x))^(n+1).
a(n) ~ 3^(1 + 3*n) * 11^(3/2 + n) / (2*sqrt((65 - 288/(1031 + 121*sqrt(73))^(1/3) + 16*(1031 + 121*sqrt(73))^(1/3)) * Pi) * n^(3/2) * (52 - (5182*2^(2/3)) / (-174721 + 65043*sqrt(73))^(1/3) + (2*(-174721 + 65043*sqrt(73)))^(1/3))^(n + 1/2)). - Vaclav Kotesovec, Nov 16 2023

A366235 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(5xA(x)).

Original entry on oeis.org

1, 1, 12, 171, 3644, 104245, 3718470, 159587365, 8014254120, 461209324905, 29936339490050, 2164061360402425, 172443226346717100, 15018744392959920925, 1419463584040707175950, 144700081009666607896125, 15826417814285141247938000, 1848740412846656456007516625

Offset: 0

Paul D. Hanna, Oct 05 2023

nonn

Related identities which hold formally for all Maclaurin series F(x):
(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.
In general, if k > 0 and e.g.f. A(x) satisfies A(x) = 1 + x*A(x) * exp(k*x*A(x)), then a(n) ~ k^n * (1 + 2*LambertW(sqrt(k)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(k)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(k)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

			E.g.f.: A(x) = 1 + x + 12*x^2/2! + 171*x^3/3! + 3644*x^4/4! + 104245*x^5/5! + 3718470*x^6/6! + 159587365*x^7/7! + 8014254120*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(5*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+4*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+3*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(+2*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(+1*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-0*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-1*x*A(x))/6! + ...
and
A(x) = 1 + 6*1*6^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 6*2*7^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 6*3*8^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 6*4*9^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 6*5*10^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...

Cf. A161633, A366232, A366233, A366234.

Cf. A365775 (dual).

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(5*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)

/* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 5^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(5*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

a(n) = n! * Sum{k=0..n} binomial(n+1, n-k)/(n+1) * 5^k * (n-k)^k / k!.
Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 5^k * (n-k)^k/k!.
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*A(x) * exp(5*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(5*x)) ).
(3) A( x/(1 + x*exp(5*x)) ) = 1 + x*exp(5*x).
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-5)*x*A(x)) for all fixed nonnegative m.
(4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-5)*x*A(x)).
(4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-4)*x*A(x)).
(4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).
(4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
(4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
(4.f) A(x) = 1 + 6 * Sum{n>=1} n*(n+5)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
a(n) ~ 5^n * (1 + 2*LambertW(sqrt(5)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(5)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(5)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A379702 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) / (1 + xexp(3x)) ).

Original entry on oeis.org

1, 0, 5, 11, 333, 2829, 78553, 1360197, 42149817, 1123709129, 40775629581, 1453036152897, 62005204699045, 2736440768515869, 135913168259011809, 7106229274104610829, 405068417020871464689, 24398077807975709138193, 1574189366334360310720405

Offset: 0

Seiichi Manyama, Dec 30 2024

nonn

Cf. A108919, A379701.

Cf. A366233.

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

a(n) = (n!/(n+1)) * Sum_{k=0..n} (2*n-3*k-1)^k * binomial(n+1,n-k)/k!.

A379847 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + xexp(3x)) ).

Original entry on oeis.org

1, 2, 17, 259, 5773, 171021, 6342937, 283094309, 14785425081, 885090944809, 59765476266061, 4494836808752049, 372655043070926821, 33769844474642217293, 3320996349535681398849, 352267766021524028011981, 40091829710459334010532593, 4873329774181782935197522641

Offset: 0

Seiichi Manyama, Jan 04 2025

nonn

Cf. A088690, A379456, A379846.

Cf. A366233, A379702.

a(n) = n!*sum(k=0, n, (4*n-3*k+1)^k*binomial(n+1, n-k)/k!)/(n+1);

a(n) = (n!/(n+1)) * Sum_{k=0..n} (4*n-3*k+1)^k * binomial(n+1,n-k)/k!.

E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(4*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A382039 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(3*x)) ).

Original entry on oeis.org

1, 1, 10, 147, 3252, 96165, 3569778, 159771717, 8378589096, 504057519945, 34227869887710, 2589957885708369, 216121694333055228, 19717935804239270013, 1952741002119283320714, 208629930642065967641805, 23919711023929511941080912, 2929406351866509691077727761

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A213644, A379690, A382040.

Cf. A366233.

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

E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*exp(3*x*A(x)).

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

A366230 Expansion of e.g.f. A(x,y) satisfying A(x,y) = 1 + xA(x,y) exp(xy A(x,y)), as a triangle read by rows.

Original entry on oeis.org

1, 1, 0, 2, 2, 0, 6, 18, 3, 0, 24, 144, 96, 4, 0, 120, 1200, 1800, 400, 5, 0, 720, 10800, 28800, 16200, 1440, 6, 0, 5040, 105840, 441000, 470400, 119070, 4704, 7, 0, 40320, 1128960, 6773760, 11760000, 6021120, 762048, 14336, 8, 0, 362880, 13063680, 106686720, 274337280, 238140000, 65028096, 4408992, 41472, 9, 0

Offset: 0

Paul D. Hanna, Nov 17 2023

A161633(n) = Sum_{k=0..n} T(n,k) for n >= 0.
A366232(n) = Sum_{k=0..n} T(n,k) * 2^k for n >= 0.
A366233(n) = Sum_{k=0..n} T(n,k) * 3^k for n >= 0.
A366234(n) = Sum_{k=0..n} T(n,k) * 4^k for n >= 0.
A366235(n) = Sum_{k=0..n} T(n,k) * 5^k for n >= 0.

			E.g.f. A(x,y) = 1 + x + (2*y + 2)*x^2/2! + (3*y^2 + 18*y + 6)*x^3/3! + (4*y^3 + 96*y^2 + 144*y + 24)*x^4/4! + (5*y^4 + 400*y^3 + 1800*y^2 + 1200*y + 120)*x^5/5! + (6*y^5 + 1440*y^4 + 16200*y^3 + 28800*y^2 + 10800*y + 720)*x^6/6! + (7*y^6 + 4704*y^5 + 119070*y^4 + 470400*y^3 + 441000*y^2 + 105840*y + 5040)*x^7/7! + (8*y^7 + 14336*y^6 + 762048*y^5 + 6021120*y^4 + 11760000*y^3 + 6773760*y^2 + 1128960*y + 40320)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins
 1;
 1, 0;
 2, 2, 0;
 6, 18, 3, 0;
 24, 144, 96, 4, 0;
 120, 1200, 1800, 400, 5, 0;
 720, 10800, 28800, 16200, 1440, 6, 0;
 5040, 105840, 441000, 470400, 119070, 4704, 7, 0;
 40320, 1128960, 6773760, 11760000, 6021120, 762048, 14336, 8, 0;
 362880, 13063680, 106686720, 274337280, 238140000, 65028096, 4408992, 41472, 9, 0;
 ...

Cf. A161633, A366232, A366233, A366234, A366235.

{T(n,k) = n! * binomial(n+1, n-k)/(n+1) * (n-k)^k/k!}
for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

T(n,k) = n! * binomial(n+1, n-k)/(n+1) * (n-k)^k / k!.
Let A(x,y)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * y^k * (n-k)^k/k!.
E.g.f. A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} T(n,k)*y^k satisfies the following formulas.
(1) A(x,y) = 1 + x*A(x) * exp(x*y*A(x,y)).
(2) A(x,y) = (1/x) * Series_Reversion( x/(1 + x*exp(x*y)) ).
(3) A( x/(1 + x*exp(x*y)), y) = 1 + x*exp(x*y).
(4) A(x,y) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+m-y)*x*A(x,y)) for all fixed nonnegative m.
(4.a) A(x,y) = 1 + Sum{n>=1} n^(n-1) * x^n/n! * A(x,y)^n * exp(-(n-y)*x*A(x)).
(4.b) A(x,y) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+1-y)*x*A(x,y)).
(4.c) A(x,y) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+2-y)*x*A(x,y)).
(4.d) A(x,y) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+3-y)*x*A(x,y)).
(4.e) A(x,y) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x,y)^n * exp(-(n+4-y)*x*A(x,y)).

cp's OEIS Frontend

A161633 E.g.f. satisfies A(x) = 1/(1 - x*exp(x*A(x))).

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A212917 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^3) ).

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A366232 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(2*x*A(x)).

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A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(4*x*A(x)).

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A365773 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2.

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A366235 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(5*x*A(x)).

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A379702 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) / (1 + x*exp(3*x)) ).

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A161633 E.g.f. satisfies A(x) = 1/(1 - xexp(xA(x))).

A366232 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(2xA(x)).

A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(4xA(x)).

A365773 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 3x*A(x))^2.

A366235 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(5xA(x)).

A379702 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) / (1 + xexp(3x)) ).

A379847 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + xexp(3x)) ).

A382039 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(3*x)) ).

A366230 Expansion of e.g.f. A(x,y) satisfying A(x,y) = 1 + xA(x,y) exp(xy A(x,y)), as a triangle read by rows.