A366234 -id:A366234 - 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

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

A366233 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(3xA(x)).

Original entry on oeis.org

1, 1, 8, 87, 1428, 31125, 847818, 27785205, 1065267864, 46802921769, 2319200977230, 127985909702409, 7785440359004916, 517616528753919933, 37344834374921549154, 2906043724955696034285, 242627026212699695954352, 21634774261037677172406609, 2052077586846349144929739542

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 + 8*x^2/2! + 87*x^3/3! + 1428*x^4/4! + 31125*x^5/5! + 847818*x^6/6! + 27785205*x^7/7! + 1065267864*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(3*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+2*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+1*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(-0*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(-2*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-3*x*A(x))/6! + ...
and
A(x) = 1 + 4*1*4^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 4*2*5^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 4*3*6^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 4*4*7^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 4*5*8^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + 254701*x^6/6! + ... + A212917(n)*x^n/n! + ...

Cf. A161633, A366232, A366234, A366235.

Cf. A365773 (dual), A212917 (exp(x*A(x))).

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(3*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) * 3^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(3*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) * 3^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) * 3^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(3*x*A(x)).
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(3*x)) ).
(3) A( x/(1 + x*exp(3*x)) ) = 1 + x*exp(3*x).
(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-3)*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-3)*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-2)*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-1)*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*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)).
a(n) ~ 3^n * (1 + 2*LambertW(sqrt(3)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(3)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(3)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023

A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 4x*A(x))^2.

Original entry on oeis.org

1, 1, 9, 73, 625, 5681, 53945, 528697, 5307489, 54298849, 564079337, 5934390441, 63098046929, 676976915473, 7319925023897, 79684985945753, 872620958369473, 9606337027601345, 106249046704511945, 1180096759408431881, 13156993620315230001, 147193406523115480049

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 + 9*x^2 + 73*x^3 + 625*x^4 + 5681*x^5 + 53945*x^6 + 528697*x^7 + 5307489*x^8 + 54298849*x^9 + 564079337*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2
also
A(x) = 1 + 1^0*x*A(x)/(1 + (-3)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-2)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + (-1)*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 0*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 1*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 2*x*A(x))^7 + ...
and
A(x) = 1 + 5*1*5^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 5*2*6^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 5*3*7^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 5*4*8^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 5*5*9^3*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, A365773, A365775.

Cf. A366234 (dual).

{a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 4^k)}
for(n=0,30, print1(a(n),", "))

{a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 4*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) * 4^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) * 4^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 4*x)^2) ).
(3) A( x/(1 + x/(1 - 4*x)^2) ) = 1 + x/(1 - 4*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-4)*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-4)*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-3)*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-2)*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-1)*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*x*A(x))^(n+1).

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

A245265 E.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^4)).

Original entry on oeis.org

1, 1, 3, 37, 649, 15461, 471571, 17456041, 760880625, 38178439849, 2167446089251, 137359883836781, 9612722107574521, 736277501363180557, 61265207586681046131, 5503291392884323494961, 530778414439201798454881, 54706967800114521799571921, 6000952913613549583603208515

Offset: 0

Vaclav Kotesovec, Jul 15 2014

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)).

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265

Cf. A161630 (p=1), A212722 (p=2), A212917 (p=3).
Cf. A030178.
Cf. A366234 (log).

Table[Sum[n! * (1 + 4*(n-k))^(k-1)/k! * Binomial[n-1,n-k],{k,0,n}],{n,0,20}]

for(n=0,30, print1(sum(k=0,n, n!*(1 + 4*(n-k))^(k-1)/k!*binomial(n-1,n-k)), ", ")) \\ G. C. Greubel, Nov 17 2017

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

a(n) ~ n^(n-1) * (1+2*LambertW(1))^(n+1/2) / (exp(n) * (LambertW(1))^(2*n) * (4*sqrt(1+LambertW(1)))). - Vaclav Kotesovec, Jul 15 2014

A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(4*x)) ).

Original entry on oeis.org

1, 1, 12, 198, 4912, 163120, 6796224, 341366704, 20088997632, 1356164492544, 103333898644480, 8773563043734016, 821474949840482304, 84093840447771701248, 9344359942839980900352, 1120159940123276849141760, 144096985208727744665288704, 19800296439825918648654561280

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A213644, A379690, A382039.

Cf. A366234, A382031.

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

E.g.f. A(x) satisfies A(x) = 1 + x*A(x)^2*exp(4*x*A(x)).

a(n) = (1/(n+1)) * Sum_{k=0..n} (4*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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A366232 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(2*x*A(x)).

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A366233 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(3*x*A(x)).

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A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 4*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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A245265 E.g.f. satisfies: A(x) = exp(x/(1-x*A(x)^4)).

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A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(4*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)).

A366233 Expansion of e.g.f. A(x) satisfying A(x) = 1 + xA(x) exp(3xA(x)).

A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 4x*A(x))^2.

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

A382040 Expansion of e.g.f. (1/x) * Series_Reversion( x(1 - xexp(4*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.