A366232 -id:A366232 - 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

A212722 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 25, 313, 5341, 115651, 3036517, 93767185, 3330162073, 133737097411, 5992748728561, 296433923379529, 16044427276953973, 943207466055927619, 59848531677741706621, 4076826825898115406241, 296742863575079244130225

Offset: 0

Paul D. Hanna, May 25 2012

nonn

			E.g.f: A(x) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ...
such that, by definition:
log(A(x))/x = 1 + x*A(x)^2 + x^2*A(x)^4 + x^3*A(x)^6 + x^4*A(x)^8 + ...
Related expansions:
log(A(x)) = x/(1-x*A(x)^2) = x + 2*x^2/2! + 18*x^3/3! + 216*x^4/4! + 3640*x^5/5! + 78000*x^6/6! + 2032464*x^7/7! + 62400128*x^8/8! + ... + n*A366232(n-1)*x^n/n! + ...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 68*x^3/3! + 880*x^4/4! + 15312*x^5/5! + 336064*x^6/6! +...
A(x)^4 = 1 + 4*x + 24*x^2/2! + 232*x^3/3! + 3232*x^4/4! + 59104*x^5/5! + 1343296*x^6/6! +...

G. C. Greubel, Table of n, a(n) for n = 0..358
Vaclav Kotesovec, Asymptotic of sequences A161630, A212722, A212917 and A245265

Cf. A161630, A212917, A245265.

Cf. A366232 (log).

Table[Sum[n! * (1 + 2*(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+2*(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^2+x*O(x^n)))); n!*polcoeff(A^m, n)}
for(n=0,21,print1(a(n),", "))

a(n) = Sum_{k=0..n} n! * (1 + 2*(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 + 2*(n-k))^(k-1)/k! * C(n-1,n-k).
a(n) ~ n^(n-1) * (1+1/(2*c))^(n+1/2) / (2*sqrt(1+c) * exp(n) * c^n), where c = LambertW(1/sqrt(2)) = 0.450600515864833072257... . - Vaclav Kotesovec, Jul 15 2014

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

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

A365772 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 2x*A(x))^2.

Original entry on oeis.org

1, 1, 5, 25, 137, 801, 4893, 30857, 199377, 1313089, 8782389, 59491257, 407308377, 2814044897, 19594237133, 137364464681, 968743846561, 6868059398273, 48921561805413, 349942779608153, 2512722402972457, 18104571857859233, 130856263145140861, 948520413875412681

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 + 5*x^2 + 25*x^3 + 137*x^4 + 801*x^5 + 4893*x^6 + 30857*x^7 + 199377*x^8 + 1313089*x^9 + 8782389*x^10 + ...
where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 2*x*A(x))^2
also
A(x) = 1 + x*A(x)/(1 + (-1)*x*A(x))^2 + 2*x^2*A(x)^2/(1 + 0*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 + 2*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 3*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 4*x*A(x))^7 + ...
and
A(x) = 1 + 3*1*3^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 3*2*4^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 3*3*5^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 3*4*6^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 3*5*7^3*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, A365773, A365774, A365775.

Cf. A366232 (dual).

nmax = 30; A[] = 0; Do[A[x] = 1 + x*A[x]/(1 - 2*x*A[x])^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 05 2023 *)

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

{a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 2*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) * 2^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) * 2^k.
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + x*A(x)/(1 - 2*x*A(x))^2.
(2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 2*x)^2) ).
(3) A( x/(1 + x/(1 - 2*x)^2) ) = 1 + x/(1 - 2*x)^2.
(4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-2)*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-2)*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-1)*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*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).
a(n) ~ 7^(n + 3/2) * sqrt(3/((-1916 + (1833997600 - 95194848*sqrt(69))^(1/3) + 2^(5/3)*(57312425 + 2974839*sqrt(69))^(1/3))*Pi)) / (2 * n^(3/2) * (1 - 53*(2/(3*(-45 + 161*sqrt(69))))^(1/3) + ((-45 + 161*sqrt(69))/2)^(1/3)/3^(2/3))^n). - Vaclav Kotesovec, Oct 05 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

A379701 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) / (1 + xexp(2x)) ).

Original entry on oeis.org

1, 0, 3, 2, 113, 304, 13747, 83600, 3590337, 38193920, 1650383171, 26535997696, 1186785903217, 26244849422336, 1234578346302771, 35176362803984384, 1757110507998276353, 61533880908307038208, 3281634015502670522371, 136392534106346468999168

Offset: 0

Seiichi Manyama, Dec 30 2024

nonn

Cf. A108919, A379702.

Cf. A366232.

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

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

A379846 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + xexp(2x)) ).

Original entry on oeis.org

1, 2, 15, 202, 3993, 104896, 3449431, 136490768, 6319722513, 335372124160, 20074806151551, 1338341234648320, 98356732036224745, 7900673166769620992, 688709957632464564231, 64754459774124307019776, 6532479591772426224737697, 703834470938326183482621952

Offset: 0

Seiichi Manyama, Jan 04 2025

nonn

Cf. A088690, A379456, A379847.

Cf. A366232, A379701.

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

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

E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(3*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A382043 E.g.f. A(x) satisfies A(x) = 1 + xA(x)^3exp(2xA(x)).

Original entry on oeis.org

1, 1, 10, 168, 4280, 146840, 6354432, 332467072, 20419261312, 1440559380096, 114820434103040, 10205253450850304, 1000815286620229632, 107355373421379825664, 12504295470535952613376, 1571670041412254073323520, 212035122185327799251468288, 30561822671438790519426154496

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A366232, A379690, A382044.

Cf. A364984, A382030.

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

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

A382044 E.g.f. A(x) satisfies A(x) = 1 + xA(x)^4exp(2xA(x)).

Original entry on oeis.org

1, 1, 12, 252, 8096, 352120, 19372512, 1290832480, 101078857728, 9098805892608, 925857411706880, 105098610198360064, 13167689873652178944, 1804954814456584081408, 268702350796640969736192, 43172786067215188056023040, 7446421094705349321120677888, 1372319952106065844255081037824

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A366232, A379690, A382043.

Cf. A365175, A382031.

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

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

cp's OEIS Frontend

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

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A212722 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^2) ).

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

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A365772 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 2*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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A379701 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) / (1 + x*exp(2*x)) ).

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

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

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

A365772 Expansion of g.f. A(x) satisfying A(x) = 1 + xA(x)/(1 - 2x*A(x))^2.

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

A379701 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) / (1 + xexp(2x)) ).

A379846 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + xexp(2x)) ).

A382043 E.g.f. A(x) satisfies A(x) = 1 + xA(x)^3exp(2xA(x)).

A382044 E.g.f. A(x) satisfies A(x) = 1 + xA(x)^4exp(2xA(x)).