A361306 -id:A361306 - cp's OEIS frontend

A214372 G.f. satisfies A(x) = x + A(x)^2*(1 + A(x))^2.

1, 1, 4, 16, 74, 364, 1876, 9993, 54582, 304040, 1720576, 9864114, 57169168, 334404892, 1971641096, 11705119157, 69911438676, 419798442416, 2532791105844, 15346498242460, 93344296450020, 569741502841020, 3488539758461880, 21422341792366320

Offset: 1

Paul D. Hanna, Jul 14 2012

nonn

			G.f.: A(x) = x + x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 +...
Related expansions:
A(x) = x + x^2*(1+x)^2 + d/dx x^4*(1+x)^4/2! + d^2/dx^2 x^6*(1+x)^6/3! + d^3/dx^3 x^8*(1+x)^8/4! +...
log(A(x)/x) = x*(1+x)^2 + d/dx x^3*(1+x)^4/2! + d^2/dx^2 x^5*(1+x)^6/3! + d^3/dx^3 x^7*(1+x)^8/4! +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 40*x^5 + 196*x^6 + 1004*x^7 + 5328*x^8 +...
(1+A(x))^2 = 1 + 2*x + 3*x^2 + 10*x^3 + 41*x^4 + 188*x^5 + 924*x^6 + 4756*x^7 + 25314*x^8 +...
Series reversion shows a relation to the Catalan numbers (A000108):
Series_Reversion( sqrt(A(x) - x) ) =  (sqrt(1+4*x) - 1)/2 - x^2, which begins
x - 2*x^2 + 2*x^3 - 5*x^4 + 14*x^5 - 42*x^6 + 132*x^7 - 429*x^8 +...
where
sqrt(A(x) - x) = x + 2*x^2 + 6*x^3 + 25*x^4 + 114*x^5 + 560*x^6 + 2880*x^7 +...+ A229042(n)*x^n +...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A229042, A361305, A361306.

a:= n-> coeff(series(RootOf(A=x+A^2*(1+A)^2, A), x, n+1), x, n):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 13 2017

Rest[CoefficientList[InverseSeries[Series[x - x^2*(1+x)^2,{x,0,20}],x],x]] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=polcoeff(serreverse(x-x^2*(1+x)^2+x*O(x^n)), n)}
for(n=1, 30, 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, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); 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, x^(2*m-1)*(1+x+x*O(x^n))^(2*m)/m!))); polcoeff(A, n)}

{a(n)=polcoeff(x + serreverse( (sqrt(1+4*x +x*O(x^n)) - 1)/2 - x^2 )^2,n)}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion( x - x^2*(1+x)^2 ).
(2) A(x) = x + Series_Reversion( (sqrt(1+4*x) - 1)/2 - x^2 )^2. - Paul D. Hanna, Oct 24 2013
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*(1+x)^(2*n)/n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1+x)^(2*n)/n! ).
Recurrence: 31*(n-2)*(n-1)*n*(104*n - 293)*a(n) = 8*(n-2)*(n-1)*(1768*n^2 - 7633*n + 7425)*a(n-1) + 20*(n-2)*(2080*n^3 - 14180*n^2 + 31612*n - 22995)*a(n-2) + 8*(2*n-5)*(4*n-13)*(4*n-11)*(104*n - 189)*a(n-3). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ 1/312*sqrt(78)*sqrt((26533 + 50583*sqrt(78))^(2/3) - 5837 + 13*(26533 + 50583*sqrt(78))^(1/3))/((26533 + 50583*sqrt(78))^(1/6)) * (4/93*((209773 + 4836*sqrt(78))^(2/3) + 3481 + 34*(209773 + 4836* sqrt(78))^(1/3))/(209773 + 4836*sqrt(78))^(1/3))^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013
a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(2*k,n-k)/(n+k+1). - Seiichi Manyama, Aug 24 2023

A361305 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^3.

Original entry on oeis.org

1, 1, 5, 23, 123, 700, 4170, 25677, 162101, 1043603, 6825429, 45222437, 302892681, 2047499720, 13950769772, 95710823820, 660609751890, 4584018016679, 31960334260971, 223782306725768, 1572921720684820, 11094267854522250, 78499108540111380, 557041048588402170

Offset: 1

Paul D. Hanna, Mar 08 2023

nonn

			G.f.: A(x) = x + x^2 + 5*x^3 + 23*x^4 + 123*x^5 + 700*x^6 + 4170*x^7 + 25677*x^8 + 162101*x^9 + 1043603*x^10 + ...
such that A(x) = x + A(x)^2 * (1 + A(x))^3.
Related series.
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 56*x^5 + 317*x^6 + 1876*x^7 + 11499*x^8 + 72352*x^9 + 464585*x^10 + ...
(1 + A(x))^3 = 1 + 3*x + 6*x^2 + 22*x^3 + 105*x^4 + 555*x^5 + 3151*x^6 + 18735*x^7 + 115200*x^8 + 726530*x^9 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..500

Cf. A361304, A361306, A214372.

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

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x) = Series_Reversion( x - x^2*(1+x)^3 ).
(2) A(x) = x + A(x)^2*(1 + A(x))^3.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * (1+x)^(3*n) / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(3*n) / n! ).
(5) A(x) = x + Series_Reversion( Series_Reversion( x*(1+x)^(3/2) ) - x^2 )^2.
From Vaclav Kotesovec, Mar 09 2023: (Start)
Recurrence: 283*(n-3)*(n-2)*(n-1)*n*(3869140*n^3 - 39431172*n^2 + 133221959*n - 149076999)*a(n) = 4*(n-3)*(n-2)*(n-1)*(1199433400*n^4 - 14022813420*n^3 + 59620648652*n^2 - 107988096753*n + 68872774500)*a(n-1) + 6*(n-3)*(n-2)*(3718243540*n^5 - 52766330452*n^4 + 294066223701*n^3 - 803084308634*n^2 + 1072900001465*n - 559958090580)*a(n-2) + 12*(n-3)*(2453034760*n^6 - 43397123748*n^5 + 316599139024*n^4 - 1218191215329*n^3 + 2605017314614*n^2 - 2932345787601*n + 1355713586640)*a(n-3) + 5*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(3869140*n^3 - 27823752*n^2 + 65967035*n - 51417072)*a(n-4).
a(n) ~ 1/(2 * sqrt(Pi*(1 + s)*(1 + 8*s + 10*s^2)) * n^(3/2) * r^(n - 1/2)), where r = 0.1321273811013026086255933373480102325835852282463... and s = 0.2180852364825231879900920777342190033594997222087... are real roots of the system of equations r + s^2 * (1+s)^3 = s, s * (1+s)^2 * (2+5*s) = 1. (End)
a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(3*k,n-k)/(n+k+1). - Seiichi Manyama, Aug 24 2023

A361304 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2n) (1 + x)^(4*n) / n!.

Original entry on oeis.org

1, 2, 18, 124, 930, 7146, 55804, 441312, 3521898, 28307510, 228820086, 1858240956, 15149110912, 123905220292, 1016261712240, 8355494725376, 68842600563918, 568266625104498, 4698576694639306, 38906632384471820, 322596353513983626, 2678048134387075560

Offset: 0

Paul D. Hanna, Mar 08 2023

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 124*x^3 + 930*x^4 + 7146*x^5 + 55804*x^6 + 441312*x^7 + 3521898*x^8 + 28307510*x^9 + ...

Cf. A361306, A214372.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); A = sum(m=0, n, Dx(m, x^(2*m)*(1+x +O(x^(n+1)))^(4*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); A = deriv( serreverse(x - x^2*(1+x +O(x^(n+3)))^4 )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*(1 + x)^4).
(3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(4*n) / n! ) is the g.f. of A361306.
(4) a(n) = (n+1) * A361306(n+1) for n >= 0.

A367234 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x))^4.

Original entry on oeis.org

1, 1, 6, 35, 226, 1561, 11276, 84150, 643730, 5021038, 39781858, 319282210, 2590312872, 21208628405, 175024439504, 1454329099044, 12157356271998, 102170610282040, 862721635191860, 7315768816166027, 62274763166575410, 531950072655682896, 4558282056420235664

Offset: 0

Seiichi Manyama, Nov 11 2023

nonn

Index entries for reversions of series

Cf. A321798, A365114, A367235.
Cf. A001003, A011270, A365150.
Cf. A361306, A378567.

a(n, s=4, t=2, u=1) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+1));

If g.f. satisfies A(x) = 1 + x*A(x)^t / (1 - x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(n+(s-1)*k-1,n-k) / (t*k+u*(n-k)+1).
From Seiichi Manyama, Dec 01 2024: (Start)
G.f.: exp( Sum_{k>=1} A378567(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - x/(1 - x)^4)^(n+1).
G.f.: (1/x) * Series_Reversion( x*(1 - x/(1 - x)^4) ). (End)

cp's OEIS Frontend

A214372 G.f. satisfies A(x) = x + A(x)^2*(1 + A(x))^2.

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A361305 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^3.

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A361304 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(4*n) / n!.

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A367234 G.f. satisfies A(x) = 1 + x*A(x)^2 / (1 - x*A(x))^4.

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A361304 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2n) (1 + x)^(4*n) / n!.

A367234 G.f. satisfies A(x) = 1 + xA(x)^2 / (1 - xA(x))^4.