A161629 -id:A161629 - cp's OEIS frontend

A213643 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x)).

1, 2, 18, 252, 4940, 124350, 3823722, 138915560, 5822192952, 276522143130, 14677209803630, 860990013672492, 55315008281020644, 3862656545279925302, 291301089508829138130, 23595204076694940812880, 2042970533426395737658352, 188298566037963463789282482

Offset: 1

Paul D. Hanna, Jun 17 2012

nonn

			E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 252*x^4/4! + 4940*x^5/5! +...
where A(x - x^2*exp(x)) = x and A(x) = x + A(x)^2*exp(A(x)).
Related expansions:
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 168*x^4/4! + 3240*x^5/5! + 80880*x^6/6! +...
A(x) = x*Catalan(x*G(x)) where G(x) = exp(A(x)):
exp(A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 349*x^4/4! + 6821*x^5/5! + 171421*x^6/6! +..., which is the e.g.f. of A161629.
A(x) = x + exp(x)*x^2 + d/dx exp(2*x)*x^4/2! + d^2/dx^2 exp(3*x)*x^6/3! + d^3/dx^3 exp(4*x)*x^8/4! +...
log(A(x)/x) = exp(x)*x + d/dx exp(2*x)*x^3/2! + d^2/dx^2 exp(3*x)*x^5/3! + d^3/dx^3 exp(4*x)*x^7/4! +...
Ordinary Generating Function:
O.g.f.: x + 2*x^2 + 18*x^3 + 252*x^4 + 4940*x^5 + 124350*x^6 +...
O.g.f.: x + 2*x^2/(1-x)^3 + 6*2!*x^3/(1-2*x)^5 + 20*3!*x^4/(1-3*x)^7 + 70*4!*x^5/(1-4*x)^9 + 252*5!*x^6/(1-5*x)^11 +...+ (2*n)!/n!*x^(n+1)/(1-n*x)^(2*n+1) +...

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A213644, A161629, A200319, A215003.

a:= n-> n!*coeff(series(RootOf(A=x+A^2*exp(A), A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 18 2013

Flatten[{1,Table[Sum[k^(n-k-1)/(n-k-1)!*(n+k-1)!/k!,{k,0,n-1}],{n,2,20}]}] (* Vaclav Kotesovec, Jul 13 2013 *)

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

{a(n)=n!*polcoeff(serreverse(x-x^2*exp(x+x*O(x^n))),n)}
for(n=1,25,print1(a(n),", "))

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, exp(m*x+x*O(x^n))*x^(2*m)/m!)); n!*polcoeff(A, n)}

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, exp(m*x+x*O(x^n))*x^(2*m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}

/* O.g.f.: */
{a(n)=polcoeff(sum(m=0,n,(2*m)!/m!*x^(m+1)/(1-m*x+x*O(x^n))^(2*m+1)),n)}

E.g.f.: A(x) = log(G(x)) where G(x) = exp(x*Catalan(x*G(x))) is the e.g.f. of A161629, and Catalan(x) = (1-sqrt(1-4*x))/(2*x).
E.g.f.: Series_Reversion(x - x^2*exp(x)).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)*x^(2*n) / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) exp(n*x)*x^(2*n-1) / n! ).
O.g.f.: Sum_{n>=0} (2*n)!/n! * x^(n+1) / (1 - n*x)^(2*n+1).
a(n) = Sum_{k=0..n-1} k^(n-k-1)/(n-k-1)! * (n+k-1)!/k!.
a(n) = n*A213644(n-1).
Limit n->infinity (a(n)/n!)^(1/n) = r*(1+r)/(1-r) = 5.5854662015218413..., where r = 0.7603592340333989... is the root of the equation (1-r^2)/r^2 = exp((r-1)/r). - Vaclav Kotesovec, Jul 13 2013
a(n) ~ (1-r) * n^(n-1) * (r*(1+r)/(1-r))^n / (sqrt(r*(1+2*r-r^2))*exp(n)). - Vaclav Kotesovec, Dec 28 2013

A214689 E.g.f. satisfies: A(x) = exp( 2xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2*x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 8, 80, 1360, 32352, 989824, 37019264, 1636370432, 83464921088, 4825001503744, 311748850464768, 22263047494942720, 1741317007049007104, 148042703132414148608, 13593236125606306316288, 1340587859012582977110016, 141329649926949217139294208

Offset: 0

Paul D. Hanna, Aug 01 2012

nonn

			E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 80*x^3/3! + 1360*x^4/4! + 32352*x^5/5! +...
log(A(x))/(2*x) = 1 + x*A(x) + 2*x^2*A(x)^2 + 5*x^3*A(x)^3 + 14*x^4*A(x)^4 +...+ A000108(n)*x^n*A(x)^n +...
log(A(x))/2 = x + 2*x^2/2! + 24*x^3/3! + 408*x^4/4! + 9760*x^5/5! +...+ A214688(n)*x^n/n! +...
Ordinary Generating Function:
O.g.f.: 1 + 2*x + 8*x^2 + 80*x^3 + 1360*x^4 + 32352*x^5 + 989824*x^6 +...
O.g.f.: 1 + 2*1*x/(1-2*x) + 2*2*x^2/(1-4*x)^3 + 2*6*2!*x^3/(1-6*x)^5 + 2*20*3!*x^4/(1-8*x)^7 + 2*70*4!*x^5/(1-10*x)^9 + 2*252*5!*x^6/(1-12*x)^11 +...+ 2*(2*n-2)!/(n-1)!*x^n/(1-2*n*x)^(2*n-1) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..190

Cf. A000108, A161629, A214688.

A214689:= func< n | n le 1 select n+1 else (&+[k*Binomial(n,k)*Binomial(2*n-k,n-k)*Factorial(n-k)*2^k*(n-k+1)^(k-1)/(2*n-k): k in [0..n]]) >;
[A214689(n): n in [0..30]]; // G. C. Greubel, Mar 07 2024

Flatten[{1,Table[Sum[n!*2^k/k!*(n-k+1)^(k-1)*Binomial[2*n-k, n-k] *k/(2*n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Feb 26 2014 *)

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

{a(n, m=1)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(2*(1-sqrt(1-4*x*A))/(2*A))); n!*polcoeff(A^m, n)}

/* From o.g.f.: */
{a(n)=polcoeff(1+2*sum(m=1, n, (2*m-2)!/(m-1)!*x^m/(1-2*m*x+x*O(x^n))^(2*m-1)), n)}
for(n=0,25,print1(a(n),", "))

def A214689(n): return n+1 if n<2 else sum(k*binomial(n,k)*binomial(2*n-k,n-k)*factorial(n-k)*2^k*(n-k+1)^(k-1)/(2*n-k) for k in range(n+1))
[A214689(n) for n in range(31)] # G. C. Greubel, Mar 07 2024

E.g.f. satisfies:
(1) A(x) = exp(2*F(x)) where F(x) = x + F(x)^2*exp(2*F(x)) is the e.g.f. of A214688.
(2) A(x) = Sum_{n>=0} a(n)*x^n/n!, where
a(n) = Sum_{k=0..n} n! * (n-k+1)^(k-1)*2^k/k! * C(2*n-k,n-k)*k/(2*n-k).
(3) Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
a(n,m) = Sum_{k=0..n} (m*k*n!/(k!*(2*n-k))) * C(2*n-k,n-k) *(n-k+m)^(k-1)*2^k
...
O.g.f.: A(x) = 1 + Sum_{n>=1} 2*(2*n-2)!/(n-1)! * x^n/(1 - 2*n*x)^(2*n-1).
a(n) ~ n^(n-1) * sqrt((r*s^3*(1-6*r*s+8*r^2*s^2)) / (1 + (1-8*r-2*r^2)*s + 8*r*(-1+2*r+r^2)*s^2 + 4*r^2*(4+r)*s^3)) / (exp(n) * r^n), where s = 1.63431733254223384712742331758... is the root of the equation 1/4*log(s)*(2-s*log(s)) = (-2*s-s^2 + sqrt(2*s+5*s^2+4*s^3+s^4))/(2*s), and r = 1/4*log(s)*(2-s*log(s)) = 0.1470215075962238276175169526... - Vaclav Kotesovec, Feb 26 2014

A382029 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, 3, 31, 529, 12601, 385891, 14440567, 638576065, 32580927505, 1883889232291, 121742057314351, 8695278706372369, 680187946863332233, 57833833258995140803, 5310742450917819399751, 523793286672328763358721, 55223769332070053104438945, 6197871354601209094032190147

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A212722, A382030, A382031.
Cf. A161629, A382042.
Cf. A000108, A214688, A214689, A379690.

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

Let F(x) be the e.g.f. of A379690. F(x) = log(A(x))/x = C(x*A(x)^2).
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(2*x)) ) ).
a(n) = n! * Sum_{k=0..n-1} (2*k+1)^(n-k-1) * binomial(n+k,k)/((n+k) * (n-k-1)!) for n > 0.

A382016 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 3, 37, 901, 32141, 1502701, 86737645, 5952271977, 473117681881, 42731313784921, 4321503662185601, 483709266378568429, 59360036142346311685, 7924411424305558028757, 1143251381667547987358581, 177245340974472998607370321, 29386977237154379581209716657

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161629, A161630, A382015.

Cf. A002293, A380513.

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

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

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 5, 67, 1465, 44541, 1735681, 82527439, 4632741905, 299875704697, 21989097804961, 1801520077445331, 163092373817762137, 16168084561101716725, 1741946677697976052577, 202668693570279026375671, 25324088113475137179021601, 3382305512670022948599733233, 480858973986045019386825360577

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A161629, A382059.
Cf. A161635, A382032.
Cf. A001764, A377546, A382033.

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

Let F(x) be the e.g.f. of A377546. F(x) = log(A(x))/x = B(x*A(x))^2.
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(x))^2 ) ).
a(n) = 2 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(2*n+k,k)/((2*n+k) * (n-k-1)!) for n > 0.

A382059 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 127, 3733, 152161, 7939261, 505087843, 37920697753, 3281899787137, 321700411900441, 35227497466867531, 4262151791317099285, 564639582580738851265, 81290104199287214904037, 12637400195063381931755731, 2109868901338065949399370161, 376504852688521502050554789889

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A161629, A382058.
Cf. A364938, A382033.
Cf. A002293, A377548, A382034.

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

Let F(x) be the e.g.f. of A377548. 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 ) ).
a(n) = 3 * n! * Sum_{k=0..n-1} (k+1)^(n-k-1) * binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.

A381982 E.g.f. A(x) satisfies A(x) = exp(x) * C(xA(x)), where C(x) = 1 + xC(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 2, 11, 139, 2829, 78981, 2802163, 120667667, 6113752025, 356342305465, 23488872131871, 1727770084512495, 140302645206245701, 12466960491079733237, 1203253101643330233707, 125351056198801059896491, 14019427299278115378992049, 1675439381194882102492648305

Offset: 0

Seiichi Manyama, Mar 11 2025

nonn

Cf. A349640, A381983.

Cf. A000108, A001764, A161629, A364983.

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

Let F(x) be the e.g.f. of A364983. F(x) = C(x*A(x)) = exp( 1/2 * Sum_{k>=1} binomial(2*k,k) * (x*A(x))^k/k ).

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A001764(k)/(n-k)!.

A382015 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 3, 31, 589, 16121, 574621, 25206595, 1312188249, 79030103185, 5404390242841, 413597889825011, 35018686148243029, 3249772250267517001, 327996955065621786309, 35769289851588288786211, 4191277822883571632163121, 525144087149768803822788257, 70060367710090279786176259633

Offset: 0

Seiichi Manyama, Mar 12 2025

nonn

Cf. A161629, A161630, A382016.

Cf. A001764, A251569.

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

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

A382042 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x)^3)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 3, 37, 733, 20181, 714541, 30903769, 1579206441, 93099946249, 6219777779641, 464382363698661, 38319628830696973, 3463058939163189133, 340172205752538636933, 36087128101110502864561, 4111807211977470782285521, 500807663307856030823859729, 64931674940413564774656214513

Offset: 0

Seiichi Manyama, Mar 13 2025

nonn

Cf. A161629, A382029.

Cf. A000108, A212917, A382039.

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

Let F(x) be the e.g.f. of A382039. F(x) = log(A(x))/x = C(x*A(x)^3).
E.g.f.: A(x) = exp( Series_Reversion( x*(1 - x*exp(3*x)) ) ).
a(n) = n! * Sum_{k=0..n-1} (3*k+1)^(n-k-1) * binomial(n+k,k)/((n+k) * (n-k-1)!) for n > 0.

cp's OEIS Frontend

A213643 E.g.f. satisfies: A(x) = x + A(x)^2*exp(A(x)).

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A214689 E.g.f. satisfies: A(x) = exp( 2*x*Catalan(x*A(x)) ), where Catalan(x) = (1-sqrt(1-4*x))/(2*x) is the g.f. of A000108.

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A382029 E.g.f. A(x) satisfies A(x) = exp(x*C(x*A(x)^2)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

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A382016 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A382058 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A382059 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

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A381982 E.g.f. A(x) satisfies A(x) = exp(x) * C(x*A(x)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

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A382015 E.g.f. A(x) satisfies A(x) = exp(x*B(x*A(x))), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

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A214689 E.g.f. satisfies: A(x) = exp( 2xCatalan(xA(x)) ), where Catalan(x) = (1-sqrt(1-4x))/(2*x) is the g.f. of A000108.

A382029 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.

A382016 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

A382058 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

A382059 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))^3), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

A381982 E.g.f. A(x) satisfies A(x) = exp(x) * C(xA(x)), where C(x) = 1 + xC(x)^2 is the g.f. of A000108.

A382015 E.g.f. A(x) satisfies A(x) = exp(xB(xA(x))), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

A382042 E.g.f. A(x) satisfies A(x) = exp(xC(xA(x)^3)), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.