A181734 -id:A181734 - cp's OEIS frontend

A198953 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^3).

1, 2, 9, 56, 400, 3095, 25240, 213633, 1859006, 16527544, 149472480, 1370794835, 12718060947, 119158146283, 1125816405458, 10714275588727, 102615375322564, 988302823695146, 9565859385140272, 93000625498797314, 907782305262566776, 8892941663606408172

Offset: 0

Paul D. Hanna, Oct 31 2011

The radius of convergence of g.f. A(x) is r = 0.095007017562450871521918431664620... with A(r) = 1.6228790124092133906198298670423120590101223122... where y=A(r) satisfies 2*y^5 + 6*y^4 - 18*y^3 + 6*y^2 - 3 = 0.

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 +...
Related expansions.
A(x)^2 = 1 + 4*x + 22*x^2 + 148*x^3 + 1105*x^4 + 8798*x^5 + 73196*x^6 +...
A(x)^3 = 1 + 6*x + 39*x^2 + 284*x^3 + 2223*x^4 + 18267*x^5 + 155445*x^6 +...
A(x)^4 = 1 + 8*x + 60*x^2 + 472*x^3 + 3878*x^4 + 32948*x^5 + 287300*x^6 +...
where A(x) = 1 + x*(A(x) + A(x)^3) + x^2*A(x)^4.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^2)*x + (1 + 2^2*A(x)^2 + A(x)^4)*x^2/2 +
(1 + 3^2*A(x)^2 + 3^2*A(x)^4 + A(x)^6)*x^3/3 +
(1 + 4^2*A(x)^2 + 6^2*A(x)^4 + 4^2*A(x)^6 + A(x)^8)*x^4/4 +
(1 + 5^2*A(x)^2 + 10^2*A(x)^4 + 10^2*A(x)^6 + 5^2*A(x)^8 + A(x)^10)*x^5/5 +...
more explicitly,
log(A(x)) = 2*x + 14*x^2/2 + 122*x^3/3 + 1118*x^4/4 + 10557*x^5/5 + 101642*x^6/6 + 991916*x^7/7 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A215623, A215624, A215654, A007863, A036765, A198951, A181734, A073157, A364375.

nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF)*(1 + x*AGF^3) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 19 2013 *)

a(n):=sum((sum((binomial(2*n+2*k+2,j-k)*binomial(n+2*k,k))/(k+n+1),k,0,j))*(-1)^(n-j)*binomial(2*n-j,n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^(2*j))*x^m/m))); polcoeff(A, n)}

{a(n)=polcoeff((1/x)*serreverse( 2*x^2*(1+x) / (1 - sqrt(1 - 4*x*(1+x)^2 +x^3*O(x^n)))),n)}

{a(n)=local(A=1+x);for(i=1,n,A=(1 + x*A)*(1 + x*(A+x*O(x^n))^3));polcoeff(A,n)}

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(2*k) ).
(2) A(x) = (1/x)*Series_Reversion( 2*x^2*(1+x) / (1 - sqrt(1 - 4*x*(1+x)^2)) ).
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A073157 (Schroeder n-paths containing no FFs).
The formal inverse of g.f. A(x) is (sqrt((1-x^2)^2 + 4*x^3) - (1+x^2)) / (2*x^3).
D-finite with recurrence: 2*n*(n+1)*(2*n+1)*(1275*n^5 - 11696*n^4 + 36827*n^3 - 40618*n^2 - 5828*n + 25368)*a(n) = 6*n*(2*n - 1)*(7650*n^6 - 66351*n^5 + 183953*n^4 - 102147*n^3 - 314787*n^2 + 450754*n - 137760)*a(n-1) - 6*(n-1)*(2*n - 3)*(34425*n^6 - 281367*n^5 + 690471*n^4 - 86579*n^3 - 1831014*n^2 + 2230808*n - 685440)*a(n-2) + 6*(22950*n^8 - 279378*n^7 + 1275447*n^6 - 2461807*n^5 + 518525*n^4 + 5756973*n^3 - 9486182*n^2 + 5962912*n - 1303680)*a(n-3) - 6*(22950*n^8 - 313803*n^7 + 1633059*n^6 - 3736233*n^5 + 1886879*n^4 + 7909228*n^3 - 16107824*n^2 + 11531408*n - 2756544)*a(n-4) + 3*(n-4)*(3*n - 14)*(3*n - 7)*(1275*n^5 - 5321*n^4 + 2793*n^3 + 12437*n^2 - 16992*n + 5328)*a(n-5). - Vaclav Kotesovec, Sep 19 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 10.5255382776611313... is the root of the equation -27 + 108*d - 108*d^2 + 324*d^3 - 72*d^4 + 4*d^5 = 0 and c = 0.5321376859604656812266678970406658537671... - Vaclav Kotesovec, Sep 19 2013
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*n+2*k+2,j-k)*binomial(n+2*k,k))/(k+n+1)))*(-1)^(n-j)*binomial(2*n-j,n-j)). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1). - Seiichi Manyama, Jul 19 2023

A198888 G.f. A(x) satisfies A(x) = (1 + xA(x))(1 + x^3*A(x)^4).

Original entry on oeis.org

1, 1, 1, 2, 7, 22, 61, 172, 528, 1695, 5447, 17486, 56778, 187064, 622149, 2080325, 6990670, 23621143, 80230388, 273687898, 937072049, 3219316096, 11095261035, 38351414036, 132915860364, 461770505371, 1607875309626, 5610314558562, 19614016834508, 68696001390320, 241007011551493

Offset: 0

Paul D. Hanna, Nov 03 2011

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 61*x^6 + 172*x^7 +...
Related expansions:
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 71*x^4 + 236*x^5 + 766*x^6 +...
A(x)^5 = 1 + 5*x + 15*x^2 + 40*x^3 + 120*x^4 + 401*x^5 + 1340*x^6 +...
where A(x) = 1 + x*A(x) + x^3*A(x)^4 + x^4*A(x)^5.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^2*A(x)^3)*x + (1 + 2^2*x^2*A(x)^3 + x^4*A(x)^6)*x^2/2 +
(1 + 3^2*x^2*A(x)^3 + 3^2*x^4*A(x)^6 + x^6*A(x)^9)*x^3/3 +
(1 + 4^2*x^2*A(x)^3 + 6^2*x^4*A(x)^6 + 4^2*x^6*A(x)^9 + x^8*A(x)^12)*x^4/4 +
(1 + 5^2*x^2*A(x)^3 + 10^2*x^4*A(x)^6 + 10^2*x^6*A(x)^9 + 5^2*x^8*A(x)^12 + x^10*A(x)^15)*x^5/5 +...
Explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 76*x^5/5 + 232*x^6/6 + 743*x^7/7 + 2629*x^8/8 + 9481*x^9/9 +...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A181734, A198957, A198953, A198951, A192415, A036765.

Table[Sum[Binomial[n+k,k]*Binomial[n+k+1,n-3*k]/(n+1),{k,0,Floor[n/3]}],{n,0,20}] (* Vaclav Kotesovec, Sep 18 2013 *)

{a(n)=sum(k=0, n\3, binomial(n+k, k)*binomial(n+k+1, n-3*k))/(n+1)}

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^3*(A+x*O(x^n))^4)); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(x^2*A^3+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, n, binomial(m+j, j)^2*(x^2*A^3+x*O(x^n))^j)*(1-x^2*A^3)^(2*m+1)*x^m/m))); polcoeff(A, n, x)}

a(n) = Sum_{k=0..[n/3]} C(n+k, k)*C(n+k+1, n-3*k)/(n+1).
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x/(1+x) - x^4 ). [Corrected by Seiichi Manyama, Dec 15 2024]
(2) A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 + x)/(1 - x^3 - x^4).
(3) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*x^(2*k)*A(x)^(3*k)] * x^n/n ).
(4) A(x) = exp( Sum_{n>=1} [Sum_{k>=0} C(n+k,k)^2*x^(2*k)*A(x)^(3*k)]*(1-x^2*A(x)^3)^(2*n+1)* x^n/n ).
Recurrence: 283*(n-2)*(n-1)*n*(n+1)*(23959952*n^4 - 257205740*n^3 + 1013304652*n^2 - 1735060589*n + 1087154052)*a(n) = 4*(n-2)*(n-1)*n*(8529742912*n^5 - 95830114896*n^4 + 406564828744*n^3 - 799079033082*n^2 + 700270562579*n - 198783157747)*a(n-1) - 8*(n-2)*(n-1)*(8625582720*n^6 - 109845231840*n^5 + 557377471920*n^4 - 1435513153260*n^3 + 1966313576808*n^2 - 1346689501571*n + 355664911636)*a(n-2) + 32*(n-2)*(4216951552*n^7 - 64244492224*n^6 + 407865945256*n^5 - 1396107234938*n^4 + 2774470392903*n^3 - 3187035309382*n^2 + 1946241786026*n - 482103205479)*a(n-3) - 16*(n-2)*(1150077696*n^7 - 19246341696*n^6 + 133834520688*n^5 - 499899483140*n^4 + 1078973257808*n^3 - 1338172075263*n^2 + 875535465587*n - 229801752572)*a(n-4) + 8*(n-4)*(2*n - 5)*(4*n - 17)*(4*n - 11)*(23959952*n^4 - 161365932*n^3 + 385447144*n^2 - 384228697*n + 132152327)*a(n-5). - Vaclav Kotesovec, Sep 18 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 3.686367878047643633... is the root of the equation -256 + 768*d - 5632*d^2 + 2880*d^3 - 1424*d^4 + 283*d^5 = 0 and c = 0.73361916425726935915879240304621641469885... - Vaclav Kotesovec, Sep 18 2013

A215623 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^4).

Original entry on oeis.org

1, 2, 11, 89, 836, 8551, 92445, 1039030, 12019135, 142151324, 1711116646, 20894534324, 258195565959, 3222677162409, 40569811695707, 514520507077695, 6567611974106756, 84310605465652750, 1087798325715407703, 14098475168420865396, 183465816241394787196

Offset: 0

Paul D. Hanna, Aug 17 2012

The radius of convergence of g.f. A(x) is r = 0.0712256396327314729661274986100... with A(r) = 1.4248895273944523042559975726479124492235978714420... where y=A(r) satisfies 3*y^7 - 4*y^6 + 16*y^5 - 28*y^4 + 8*y^3 - 4 = 0.

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + 8551*x^5 + 92445*x^6 + ...
Related expansions.
A(x)^4 = 1 + 8*x + 68*x^2 + 652*x^3 + 6750*x^4 + 73544*x^5 + 831078*x^6 + ...
A(x)^5 = 1 + 10*x + 95*x^2 + 965*x^3 + 10350*x^4 + 115507*x^5 + ...
where A(x) = 1 + x*(A(x) + A(x)^4) + x^2*A(x)^5.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^3)*x + (1 + 2^2*A(x)^3 + A(x)^6)*x^2/2 +
  (1 + 3^2*A(x)^3 + 3^2*A(x)^6 + A(x)^9)*x^3/3 +
  (1 + 4^2*A(x)^3 + 6^2*A(x)^6 + 4^2*A(x)^9 + A(x)^12)*x^4/4 +
  (1 + 5^2*A(x)^3 + 10^2*A(x)^6 + 10^2*A(x)^9 + 5^2*A(x)^12 + A(x)^15)*x^5/5 + ...
more explicitly,
log(A(x)) = 2*x + 18*x^2/2 + 209*x^3/3 + 2550*x^4/4 + 32082*x^5/5 + 411705*x^6/6 + 5356416*x^7/7 + ....

Seiichi Manyama, Table of n, a(n) for n = 0..876

Cf. A198953, A215624, A036765, A198951, A181734, A364331, A364336, A364376.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^(3*j))*x^m/m))); polcoeff(A, n)}

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x*A)*(1+x*A^4)+x*O(x^n)); polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

G.f. satisfies A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(3*k)).
The formal inverse of g.f. A(x) is (sqrt((1-x^3)^2 + 4*x^4) - (1+x^3))/(2*x^4).
a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(n+3*k+1,n-k) / (n+3*k+1). - Seiichi Manyama, Jul 19 2023
From  Peter Bala, Sep 10 2024: (Start)
x/series_reversion(x*A(x)) = 1 + 2*x + 7*x^2 + 39*x^3 + 242*x^4 + 1634*x^5 + ..., the g.f. of A364336.
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 15*x^2 + 163*x^3 + 2070*x^4 + 28698*x^5 + ..., the g.f. of A364331. (End)

A215624 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 13, 130, 1518, 19358, 261323, 3670828, 53100530, 785657529, 11834135909, 180863294507, 2797643204500, 43715591710804, 689030031494554, 10941710269299893, 174889301792724294, 2811464199460768704, 45426696813655278251

Offset: 0

Paul D. Hanna, Aug 17 2012

nonn

The radius of convergence of g.f. A(x) is r = 0.05685644444171304880925020950930... with A(r) = 1.3208055627586104770123863310077013110788003146438630... where y=A(r) satisfies 4*y^9 - 5*y^8 + 25*y^6 - 40*y^5 + 10*y^4 - 5 = 0.

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 130*x^3 + 1518*x^4 + 19358*x^5 +...
Related expansions.
A(x)^5 = 1 + 10*x + 105*x^2 + 1250*x^3 + 16120*x^4 + 219162*x^5 +...
A(x)^6 = 1 + 12*x + 138*x^2 + 1720*x^3 + 22803*x^4 + 315840*x^5 +...
where A(x) = 1 + x*(A(x) + A(x)^5) + x^2*A(x)^6.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^4)*x + (1 + 2^2*A(x)^4 + A(x)^8)*x^2/2 +
  (1 + 3^2*A(x)^4 + 3^2*A(x)^8 + A(x)^12)*x^3/3 +
  (1 + 4^2*A(x)^4 + 6^2*A(x)^8 + 4^2*A(x)^12 + A(x)^16)*x^4/4 +
  (1 + 5^2*A(x)^4 + 10^2*A(x)^8 + 10^2*A(x)^12 + 5^2*A(x)^16 + A(x)^20)*x^5/5 +...
more explicitly,
log(A(x)) = 2*x + 22*x^2/2 + 320*x^3/3 + 4886*x^4/4 + 76962*x^5/5 + 1236784*x^6/6 + 20152260*x^7/7 +...

Seiichi Manyama, Table of n, a(n) for n = 0..807

Cf. A198953, A215623, A036765, A198951, A181734.

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^(4*j))*x^m/m))); polcoeff(A, n)}

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x*A)*(1+x*A^5)+x*O(x^n)); polcoeff(A, n)}
for(n=0,21,print1(a(n),", "))

G.f. satisfies A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(4*k)).
The formal inverse of g.f. A(x) is (sqrt((1-x^4)^2 + 4*x^5) - (1+x^4))/(2*x^5).
a(n) = Sum_{k=0..n} binomial(n+4*k+1,k) * binomial(n+4*k+1,n-k) / (n+4*k+1). - Seiichi Manyama, Jul 19 2023

A274379 G.f. satisfies A(x) = (1 + xA(x))^3 (1 + x^2*A(x)^3).

Original entry on oeis.org

1, 3, 13, 70, 429, 2842, 19794, 142758, 1056655, 7980280, 61251261, 476387379, 3746317414, 29738316330, 237968639936, 1917578268288, 15546796822656, 126728260011920, 1037987924978125, 8538459191677170, 70509828893263474, 584310452973463242, 4857624566855734836, 40501472981905806550, 338594135314564168494, 2837641019938074131463, 23835438376045780734390, 200633658871150345742269, 1692132786239339256115050, 14297391426538004065333910, 121009206594941545408186768

Offset: 0

Paul D. Hanna, Aug 04 2016

nonn

			 G.f.: A(x) = 1 + 3*x + 13*x^2 + 70*x^3 + 429*x^4 + 2842*x^5 + 19794*x^6 + 142758*x^7 + 1056655*x^8 + 7980280*x^9 + ...
such that A(x) = 1 + 3*x*A(x) + x^2*(3*A(x)^2 + A(x)^3) + x^3*(A(x)^3 + 3*A(x)^4) + 3*x^4*A(x)^5 + x^5*A(x)^6.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A181734, A274378.

Cf. A274735, A369600.

{a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^3 * (1 + x^2*A^3) + x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x^2*(1+x)^3)/(1+x +x^2*O(x^n) )^3 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. satisfies: A(x) = (1/x) * Series_Reversion( x*(1 - x^2*(1+x)^3) / (1+x)^3 ).
G.f. satisfies: A( x*(1 - x^2*(1+x)^3)/(1+x)^3 ) = (1+x)^3/(1 - x^2*(1+x)^3).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(3*n+3*k+3,n-2*k). - Seiichi Manyama, Jan 27 2024

A274734 G.f. satisfies A(x) = (1 + xA(x))^2 (1 + x*A(x)^2).

Original entry on oeis.org

1, 3, 15, 94, 661, 4983, 39363, 321587, 2694860, 23035341, 200068651, 1760558682, 15663027711, 140648129383, 1273083938979, 11603500739475, 106404140837773, 980977232554344, 9087285865886766, 84541177049414342, 789545725457924023, 7399515198155161271, 69568021610270590583, 655960254857760518109, 6201585037793334756198, 58775103307105512895151

Offset: 0

Paul D. Hanna, Aug 02 2016

nonn

More generally, if G(x) satisfies
G(x) = (1 + a*x*G(x))^m * (1 + b*x*G(x)^2), then
G(x) = (1/x) * Series_Reversion( x * (1 - b*x*(1 + a*x)^m) / (1 + a*x)^m ).

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 94*x^3 + 661*x^4 + 4983*x^5 + 39363*x^6 + 321587*x^7 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A181734, A274735.

{a(n) = my(A=1); for(i=1,n, A = (1 + x*A)^2 * (1 + x*A^2) + x*O(x^n) ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x*(1+x)^2)/(1+x +x^2*O(x^n) )^2 ); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

G.f.: (1/x) * Series_Reversion( x * (1 - x*(1+x)^2) / (1+x)^2 ).
Recurrence: 31*(n-1)*n*(n+1)*(5974*n^3 - 40359*n^2 + 90115*n - 67124)*a(n) = 2*(n-1)*n*(1003632*n^4 - 7282128*n^3 + 18518502*n^2 - 18822839*n + 5649607)*a(n-1) - 2*(n-1)*(740776*n^5 - 6486068*n^4 + 21715762*n^3 - 34616651*n^2 + 26123385*n - 7413210)*a(n-2) + 2*(2*n - 5)*(65714*n^5 - 575377*n^4 + 1957337*n^3 - 3264653*n^2 + 2726129*n - 941430)*a(n-3) + 4*(n-3)*(2*n - 7)*(2*n - 5)*(5974*n^3 - 22437*n^2 + 27319*n - 11394)*a(n-4). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ sqrt((s*(1+r*s)*(2 + s + 3*r*s^2)) / (1 + r*(1 + 6*s*(1+r*s)))) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.099424837262345547872398211374352678... and s = 2.183663565361369673488934371066403742... are roots of the system of equations (1 + r*s)^2*(1 + r*s^2) = s, 2*r*(1 + s + 2*r^2*s^3 + r*s*(1 + 3*s)) = 1. - Vaclav Kotesovec, Nov 18 2017
a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(2*n+2*k+2,n-k). - Seiichi Manyama, Jan 27 2024

A274378 G.f. satisfies A(x) = (1 + xA(x))^2 (1 + x^2*A(x)^3).

Original entry on oeis.org

1, 2, 6, 24, 111, 552, 2873, 15458, 85312, 480314, 2747845, 15928080, 93347153, 552181372, 3292571913, 19769887128, 119430685503, 725375643416, 4426786390959, 27131644746326, 166932630227613, 1030684209393288, 6383992918008611, 39657230694169284, 247008096338698523, 1542292860296588558, 9651791500807437834, 60528789932966226468, 380333245334293851637, 2394179659042901060436, 15096873553004201457425

Offset: 0

Paul D. Hanna, Aug 04 2016

nonn

			 G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 111*x^4 + 552*x^5 + 2873*x^6 + 15458*x^7 + 85312*x^8 +...
such that A(x) = 1 + 2*x*A(x) + x^2*(A(x)^2 + A(x)^3) + 2*x^3*A(x)^4 + x^4*A(x)^5.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A181734, A274379.

{a(n) = my(A=1); for(i=1, n, A = (1 + x*A)^2 * (1 + x^2*A^3) + x*O(x^n) ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x^2*(1+x)^2)/(1+x +x^2*O(x^n) )^2 ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

G.f. satisfies: A(x) = (1/x) * Series_Reversion( x*(1 - x^2*(1+x)^2) / (1+x)^2 ).
G.f. satisfies: A( x*(1 - x^2*(1+x)^2)/(1+x)^2 ) = (1+x)^2/(1 - x^2*(1+x)^2).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+k,k) * binomial(2*n+2*k+2,n-2*k). - Seiichi Manyama, Jan 27 2024

A379080 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^2)^2 ).

Original entry on oeis.org

1, 2, 7, 32, 163, 886, 5039, 29616, 178446, 1096356, 6842452, 43259122, 276462247, 1783114592, 11591769207, 75874998822, 499643588823, 3307746965238, 22001986381873, 146972401234478, 985535271867577, 6631547191254298, 44763982636889092, 303037237861086682

Offset: 0

Seiichi Manyama, Dec 15 2024

nonn

Cf. A181734, A379081.

Cf. A200719, A379084.

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

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379084(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^2*A(x)^(5/2)) )^2.
(3) A(x) = B(x)^2 where B(x) is the g.f. of A200719.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^2 )^(2*(n+1)).
a(n) = 2 * Sum_{k=0..floor(n/2)} binomial(2*n+k+2,k) * binomial(2*n+k+2,n-2*k)/(2*n+k+2) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(2*n+k+2,n-2*k).

A379081 Expansion of (1/x) * Series_Reversion( x * (1/(1 + x) - x^2)^3 ).

Original entry on oeis.org

1, 3, 15, 94, 657, 4905, 38299, 308928, 2554092, 21528728, 184318944, 1598427531, 14011401996, 123946608699, 1105090991634, 9920335032821, 89589290332200, 813367589142888, 7419376746340780, 67965042988027335, 624971955439306953, 5766825797557702751, 53380176096582823851

Offset: 0

Seiichi Manyama, Dec 15 2024

nonn

Cf. A181734, A379080.

Cf. A379086, A379088.

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

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{k>=1} A379086(k) * x^k/k ).
(2) A(x) = ( (1 + x*A(x)) * (1 + x^2*A(x)^(7/3)) )^3.
(3) A(x) = B(x)^3 where B(x) is the g.f. of A379088.
a(n) = (1/(n+1)) * [x^n] 1/( 1/(1 + x) - x^2 )^(3*(n+1)).
a(n) = 3 * Sum_{k=0..floor(n/2)} binomial(3*n+k+3,k) * binomial(3*n+k+3,n-2*k)/(3*n+k+3) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+k+2,k) * binomial(3*n+k+3,n-2*k).

cp's OEIS Frontend

A198953 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^3).

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A198888 G.f. A(x) satisfies A(x) = (1 + x*A(x))*(1 + x^3*A(x)^4).

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A215623 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^4).

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A215624 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^5).

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A274379 G.f. satisfies A(x) = (1 + x*A(x))^3 * (1 + x^2*A(x)^3).

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A274734 G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x*A(x)^2).

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A274378 G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x^2*A(x)^3).

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A198953 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^3).

A198888 G.f. A(x) satisfies A(x) = (1 + xA(x))(1 + x^3*A(x)^4).

A215623 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^4).

A215624 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^5).

A274379 G.f. satisfies A(x) = (1 + xA(x))^3 (1 + x^2*A(x)^3).

A274734 G.f. satisfies A(x) = (1 + xA(x))^2 (1 + x*A(x)^2).

A274378 G.f. satisfies A(x) = (1 + xA(x))^2 (1 + x^2*A(x)^3).