A161630 -id:A161630 - 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

A052873 E.g.f. A(x) satisfies A(x) = exp(xA(x)/(1 - xA(x))).

Original entry on oeis.org

1, 1, 5, 46, 629, 11496, 263857, 7301680, 236748969, 8806142080, 369714769181, 17296339048704, 892335712777885, 50333180563864576, 3081739132775658825, 203555129140352505856, 14428195498061848405073, 1092403962489972428144640, 87990832863810814525250869

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

Seiichi Manyama, Table of n, a(n) for n = 0..357
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 844.
R. Lorentz, S. Tringali, C. H. Yan, Generalized Goncarov polynomials, arXiv preprint arXiv:1511.04039 [math.CO], 2015.

Cf. A052868, A088690, A161630.

Cf. A382036, A382037, A382038.

spec := [S,{C=Sequence(B,1 <= card),S=Set(C),B=Prod(Z,S)},labeled]:
seq(combstruct[count](spec,size=n), n=0..20);
# Alternatively:
a := n -> `if`(n=0,1, n!*hypergeom([1-n],[2],-n-1)):
seq(simplify(a(n)), n=0..16); # Peter Luschny, Apr 20 2016

Table[Sum[(n+1)^(k-1)*n!/k!*Binomial[n-1,k-1],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 08 2014 *)

{a(n)=if(n==0,1,sum(k=0,n,(n+1)^(k-1)*n!/k!*binomial(n-1,k-1)))} \\ Paul D. Hanna, Sep 08 2012

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,(m+1)^(m-1)*x^m/m!/(1-x*A+x*O(x^n))^m));n!*polcoeff(A,n)} \\ Paul D. Hanna, Sep 08 2012

E.g.f.: exp(RootOf(exp(_Z)*x*_Z+exp(_Z)*x-_Z)).
1 = Sum_{n>=0} a(n)*exp((n+1)*x/(x-1))*x^n/n!. - Vladeta Jovovic, Jul 20 2005
a(n) = Sum_{k=0..n} (n+1)^(k-1)*n!/k!*binomial(n-1,k-1). - Vladeta Jovovic, Jul 02 2006
E.g.f. satisfies: A(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n! / (1-x*A(x))^n. - Paul D. Hanna, Sep 08 2012
Equivalently:
E.g.f. satisfies: A(x) = exp( x*A(x)/(1 - x*A(x)) ). - Olivier Gérard, Dec 29 2013
a(n) ~ (sqrt(5)-1) * 2^(n-1/2) * n^(n-1) * exp((sqrt(5)-1 + (sqrt(5)-3)*n)/2) / (5^(1/4) * (3-sqrt(5))^(n+1/2)). - Vaclav Kotesovec, Jan 08 2014
a(n) = n!*hypergeom([1-n],[2],-n-1) for n >= 1. - Peter Luschny, Apr 20 2016
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x) ) ). - Seiichi Manyama, Mar 15 2025

New name using e.g.f., Vaclav Kotesovec, Jan 08 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

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

A161635 E.g.f. satisfies A(x) = exp( x/(1 - x*A(x))^2 ).

Original entry on oeis.org

1, 1, 5, 43, 553, 9501, 204961, 5330599, 162432593, 5677941817, 224018814241, 9848702243931, 477481361216377, 25309471236379669, 1456206709854725921, 90387017392004356591, 6020486941130334199201, 428348710658269120403313

Offset: 0

Paul D. Hanna, Jun 19 2009

nonn

			E.g.f: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 553*x^4/4! + 9501*x^5/5! +...
log(A(x))/x = 1 + 2*x*A(x) + 3*x^2*A(x)^2 + 4*x^3*A(x)^3 + 5*x^4*A(x)^4 +...

Seiichi Manyama, Table of n, a(n) for n = 0..360

Cf. A161630, A364980.

Flatten[{1,Table[n!*Sum[(n-k+1)^(k-1)/k! * Binomial[n+k-1,n-k],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jan 10 2014 *)
FindRoot[{2*s*r^2 == (1-r*s)^3, r == Log[s]*(1-r*s)^2},{r,1/2},{s,1}, WorkingPrecision->50] (* program for numerical values of constants r and s, Vaclav Kotesovec, Jan 10 2014 *)

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

{a(n,m=1)=my(A=1+x+x*O(x^n));for(i=1,n,A=exp(x/(1-x*A)^2));n!*polcoeff(A^m,n)}

a(n) = n!*Sum_{k=0..n} (n-k+1)^(k-1)/k! * C(n+k-1,n-k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = n!*Sum_{k=0..n} m*(n-k+m)^(k-1)/k! * C(n+k-1,n-k).
...
E.g.f.: A(x) = (1/x)*Series_Reversion[ (1-x)^2*LambertW(x/(1-x)^2) ].
a(n) ~ sqrt(s*(1+r*s)/(3+2*r-6*r*s+3*r^2*s^2)) * n^(n-1) / (exp(n) * r^n), where r = 0.219092370374602436272454670371..., s = 1.952248277910295452167538973654... are the roots of the equations 2*s*r^2 = (1-r*s)^3 and r = log(s) * (1-r*s)^2. - Vaclav Kotesovec, Jan 10 2014

A361142 E.g.f. satisfies A(x) = exp( xA(x)^2/(1 - xA(x)) ).

Original entry on oeis.org

1, 1, 7, 91, 1773, 46401, 1529593, 60911103, 2845757449, 152663425633, 9250206248781, 624880915165959, 46569571425664477, 3795729136868379777, 335902071304953561073, 32074779600414913885231, 3287242849289861637185937, 359917016243351870997841473

Offset: 0

Seiichi Manyama, Mar 02 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..338

Cf. A000262, A361143.

Cf. A052868, A052873, A161630.

Table[n! * Sum[(n+k+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 03 2023 *)

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

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

a(n) ~ s^2 * sqrt((2 - r*s)/(2 + r*s*(-2 + s*(2 - r*s)^2))) * n^(n-1) / (exp(n) * r^(n - 1/2)), where r = 0.14220768719194290600038416000340972911571484385125... and s = 1.549730657609106944767484487465870359529391502493... are roots of the system of equations exp(r*s^2/(1 - r*s)) = s, r*s^2*(2 - r*s) = (1 - r*s)^2. - Vaclav Kotesovec, Mar 03 2023

A382033 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^3), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 7, 109, 2653, 88261, 3731581, 191571493, 11576241769, 804996352873, 63324553740121, 5559962513556001, 539015912053933645, 57188111522488589293, 6591136171961660099509, 820029701725988751533341, 109537705061927547203868241, 15635869913619342121140932689

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161630, A382032, A382034.

Cf. A001764, A377554.

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

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(3*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377554. F(x) = log(A(x))/x = B(x*A(x))^3.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^3 ) ).

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

A382032 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x))^2), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 5, 55, 937, 21741, 639841, 22839139, 958882289, 46304377849, 2528571710881, 154076164781991, 10364272238514217, 762867688235619877, 60989719558159065857, 5263030218009265964011, 487578723768665716788961, 48266847740986728218648433, 5084697384633390178057209793

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161630, A382033, A382034.

Cf. A000108, A377553, A382036.

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

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377553. F(x) = log(A(x))/x = C(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^2 ) ).

A382034 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 181, 5713, 246881, 13570081, 906180997, 71250724833, 6448375469665, 660286026034561, 75472025139452261, 9525947428687403473, 1315935073971181422721, 197485196722573989608289, 31993978774204625549549221, 5565216938342017912128576961, 1034506012356981473110554574145

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161630, A382032, A382033.

Cf. A002293, A377630.

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

a(n) = (n-1)! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(4*n,k)/(n-k-1)! for n > 0.
Let F(x) be the e.g.f. of A377630. F(x) = log(A(x))/x = B(x*A(x))^4.
E.g.f.: A(x) = exp( Series_Reversion( x/(1 + x*exp(x))^4 ) ).

cp's OEIS Frontend

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

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A052873 E.g.f. A(x) satisfies A(x) = exp(x*A(x)/(1 - 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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A212917 E.g.f. satisfies: A(x) = exp( x/(1 - x*A(x)^3) ).

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A161635 E.g.f. satisfies A(x) = exp( x/(1 - x*A(x))^2 ).

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A361142 E.g.f. satisfies A(x) = exp( x*A(x)^2/(1 - x*A(x)) ).

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PARI

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A382033 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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PARI

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

A052873 E.g.f. A(x) satisfies A(x) = exp(xA(x)/(1 - xA(x))).

A361142 E.g.f. satisfies A(x) = exp( xA(x)^2/(1 - xA(x)) ).

A382033 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^3), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

A382032 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x))^2), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

A382034 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^4), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.