A007863 -id:A007863 - cp's OEIS frontend

A011365 Reciprocal of g.f. for A007863.

1, -2, -3, -11, -49, -244, -1301, -7265, -41945, -248357, -1499838, -9202478, -57204263, -359491066, -2280171089, -14578108497, -93850280001, -607856133804, -3958143058643, -25897182540675, -170164359751174, -1122422557526850, -7429480035700452

Offset: 0

N. J. A. Sloane

sign

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A007863.

a(n) = if (n==0, 1, if (n==1, -2, -sum(i=0, n, binomial(n+i-2,n-2)*binomial(n+i-1,n-i))/(n-1))); \\ Michel Marcus, Dec 29 2018

a(n) = -(Sum_{i=0..n} binomial(n+i-2,n-2)*binomial(n+i-1,n-i))/(n-1), n>1; a(0) = 1, a(1) = -2. - Vladimir Kruchinin, Feb 15 2013

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

Original entry on oeis.org

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

A198951 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^3*A(x)^3).

Original entry on oeis.org

1, 1, 1, 2, 6, 16, 39, 99, 271, 763, 2146, 6062, 17359, 50337, 147057, 431874, 1275273, 3786649, 11298031, 33846202, 101762937, 306997821, 929038518, 2819426688, 8578433304, 26163061776, 79970186791, 244938841096, 751646959402, 2310683396056, 7115199919151

Offset: 0

Paul D. Hanna, Oct 31 2011

a(n) is also the number of rooted labeled trees on n nodes such that each node has 0, 1, 3, or 4 children. - Patrick Devlin, Mar 04 2012

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 16*x^5 + 39*x^6 + 99*x^7 + ...
Related expansions:
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 36*x^4 + 105*x^5 + 292*x^6 + ...
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 67*x^4 + 200*x^5 + 582*x^6 + ...
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^2*A(x)^2)*x + (1 + 2^2*x^2*A(x)^2 + x^4*A(x)^4)*x^2/2 +
(1 + 3^2*x^2*A(x)^2 + 3^2*x^4*A(x)^4 + x^6*A(x)^6)*x^3/3 +
(1 + 4^2*x^2*A(x)^2 + 6^2*x^4*A(x)^4 + 4^2*x^6*A(x)^6 + x^8*A(x)^8)*x^4/4 +
(1 + 5^2*x^2*A(x)^2 + 10^2*x^4*A(x)^4 + 10^2*x^6*A(x)^6 + 5^2*x^8*A(x)^8 + x^10*A(x)^10)*x^5/5 + ...
more explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 17*x^4/4 + 51*x^5/5 + 136*x^6/6 + 393*x^7/7 + 1233*x^8/8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..600

Cf. A198953, A007863, A036765, A186241, A200731, A228960.

a:= n-> coeff(series(RootOf(A=(1+x*A)*(1+x^3*A^3), A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2012

InverseSeries[ Series[ x/((1 + x)*(1 + x^3)), {x, 0, 31}], x] // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Sep 10 2013 *)

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

{a(n)=polcoeff((1+x+x^3+x^4+x*O(x^n))^(n+1)/(n+1), 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*j)*(A+x*O(x^n))^(2*j))*x^m/m))); polcoeff(A, n)}

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

G.f. satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(2*k) * A(x)^(2*k) ).
(2) A(x) = (1/x)*Series_Reversion(x/((1+x)*(1+x^3))).
(3) a(n) = [x^n] (1 + x + x^3 + x^4)^(n+1) / (n+1).
(4) A(x) = exp( Sum_{n>=1} x^n/n * (1-x^2*A(x)^2)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2 * x^(2*k) * A(x)^(2*k) ).
D-finite with recurrence: 3*(n+1)*(3*n+2)*(3*n+4)*(119*n^3 - 210*n^2 + 73*n - 6)*a(n) = 2*(6664*n^6 - 1764*n^5 - 11585*n^4 + 426*n^3 + 4129*n^2 - 102*n - 288)*a(n-1) - 18*(n-1)*(1190*n^5 - 910*n^4 - 1937*n^3 + 895*n^2 + 606*n - 216)*a(n-2) + 162*(n-2)*(n-1)*(2*n-3)*(119*n^3 + 147*n^2 + 10*n - 24)*a(n-3). - Vaclav Kotesovec, Sep 09 2013
a(n) ~ c*d^n/n^(3/2), where d = 1/81*((2144134+520506*sqrt(17))^(2/3)+112*(2144134+520506*sqrt(17))^(1/3)-2036)/(2144134+520506*sqrt(17))^(1/3) = 3.23407602060970245... is the root of the equation -324 + 180*d - 112*d^2 + 27*d^3 = 0 and c = 0.6286981954423757284622435... - Vaclav Kotesovec, Sep 09 2013
A(1/d) = 370/243 + (3*sqrt(17)/509 - 3070/123687)*(2144134+520506*sqrt(17))^(1/3) + (141*sqrt(17)/2072648 - 129529/503653464)*(2144134+520506*sqrt(17))^(2/3) = 2.053716618436594614948796... - Vaclav Kotesovec, Sep 10 2013
From Peter Bala, Jun 21 2015: (Start)
a(n) = 1/(n + 1)*Sum_{k = 0..floor(n/3)} binomial(n + 1,k)* binomial(n + 1,n - 3*k). Applying Maple's sumrecursion command to this formula gives the above recurrence of Kotesovec.
More generally, the coefficient of x^n in A(x)^r equals r/(n + r)*Sum_{k = 0..floor(n/3)} binomial(n + r,k)*binomial(n + r,n - 3*k) by the Lagrange-Bürmann formula.
O.g.f. A(x) = exp(Sum_{n >= 1} A228960(n)*x^n/n), where A228960(n) = Sum_{k = 0..floor(n/3)} binomial(n,k)*binomial(n,3*k). Cf. A036765, A186241 and A200731. (End)

A192415 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^3*A(x)^2).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 23, 51, 120, 286, 681, 1636, 3985, 9803, 24257, 60338, 150931, 379501, 958360, 2429294, 6179380, 15769380, 40361087, 103579221, 266471500, 687098810, 1775440421, 4596689688, 11922774513, 30977768907, 80615085087, 210103228155, 548352756656, 1433053608502

Offset: 0

Paul D. Hanna, Nov 02 2011

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 23*x^6 + 51*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 36*x^5 + 82*x^6 + 190*x^7 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 33*x^4 + 84*x^5 + 205*x^6 + 498*x^7 +...
where A(x) = 1 + x*A(x) + x^3*A(x)^2 + x^4*A(x)^3.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^2*A(x))*x + (1 + 2^2*x^2*A(x) + x^4*A(x)^2)*x^2/2 +
(1 + 3^2*x^2*A(x) + 3^2*x^4*A(x)^2 + x^6*A(x)^3)*x^3/3 +
(1 + 4^2*x^2*A(x) + 6^2*x^4*A(x)^2 + 4^2*x^6*A(x)^3 + x^8*A(x)^4)*x^4/4 +
(1 + 5^2*x^2*A(x) + 10^2*x^4*A(x)^2 + 10^2*x^6*A(x)^3 + 5^2*x^8*A(x)^4 + x^10*A(x)^5)*x^5/5 +...
Explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 31*x^5/5 + 70*x^6/6 + 176*x^7/7 + 469*x^8/8 + 1228*x^9/9 + 3161*x^10/10 +...

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

Cf. A198951, A198953, A198957, A007863, A036765, A104545.

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

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

x='x; y='y; Fxy = (1+x*y) * (1 + x^3*y^2) - y;
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy,y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(34)  \\ Gheorghe Coserea, Nov 30 2016

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(2*k) * A(x)^k ).
D-finite with recurrence: 4*(n+1)*(n+2)*(217*n^3 - 1239*n^2 + 1838*n - 336)*a(n) = 6*(n+1)*(434*n^4 - 2261*n^3 + 2339*n^2 + 1792*n - 1344)*a(n-1) - (n-1)*(2821*n^4 - 13286*n^3 + 7829*n^2 + 18464*n - 4032)*a(n-2) + 6*(868*n^5 - 6258*n^4 + 13981*n^3 - 7438*n^2 - 7769*n + 6136)*a(n-3) + 2*(n-3)*(2*n - 5)*(434*n^3 - 1393*n^2 + 211*n + 1048)*a(n-4) + 2*(n-4)*(2*n - 7)*(217*n^3 - 588*n^2 + 11*n + 480)*a(n-5). - Vaclav Kotesovec, Sep 19 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 2.730683387097269698... is the root of the equation -4 - 8*d - 24*d^2 + 13*d^3 - 12*d^4 + 4*d^5 = 0 and c = 2.078548317061344694159945441842754... is the root of the equation -1 - 67*c^2 - 19811*c^4 + 36463*c^6 - 41664*c^8 + 7936*c^10 = 0. - Vaclav Kotesovec, Sep 19 2013, updated Nov 28 2016
a(n) = Sum_{k=0..n/2+1} C(n-k+2,k-1)*C(n-k+2,2*k-1)/(n-k+2). - Vladimir Kruchinin, Feb 12 2019

A198957 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^2*A(x)^4).

Original entry on oeis.org

1, 1, 2, 7, 26, 102, 424, 1827, 8078, 36466, 167376, 778718, 3664164, 17407068, 83375616, 402198915, 1952296598, 9528757098, 46735576816, 230227356906, 1138609205372, 5651170500612, 28138939936704, 140527262919342, 703704207921932, 3532664478314484, 17775185122527776

Offset: 0

Paul D. Hanna, Nov 01 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 102*x^5 + 424*x^6 + 1827*x^7 +...
Related expansions:
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 237*x^4 + 1028*x^5 + 4570*x^6 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 375*x^4 + 1681*x^5 + 7660*x^6 +...
where A(x) = 1 + x*A(x) + x^2*A(x)^4 + x^3*A(x)^5.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A(x)^3)*x + (1 + 2^2*x*A(x)^3 + x^2*A(x)^6)*x^2/2 +
(1 + 3^2*x*A(x)^3 + 3^2*x^2*A(x)^6 + x^3*A(x)^9)*x^3/3 +
(1 + 4^2*x*A(x)^3 + 6^2*x^2*A(x)^6 + 4^2*x^3*A(x)^9 + x^4*A(x)^12)*x^4/4 +
(1 + 5^2*x*A(x)^3 + 10^2*x^2*A(x)^6 + 10^2*x^3*A(x)^9 + 5^2*x^4*A(x)^12 + x^5*A(x)^15)*x^5/5 +...
more explicitly,
log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 75*x^4/4 + 356*x^5/5 + 1746*x^6/6 + 8660*x^7/7 + 43299*x^8/8 +...
Also, g.f. A(x) = G(x*A(x)) where G(x) = A(x/G(x)) (g.f. of A104545) begins:
G(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 55*x^7 + 129*x^8 +...

Cf. A198953, A198951, A007863, A036765, A104545.

CoefficientList[1/x*InverseSeries[Series[2*x^3*(1+x)/(1 - Sqrt[1-4*x^2*(1+x)^2]), {x, 0, 20}], x],x] (* Vaclav Kotesovec, May 28 2014 *)

a(n):=sum(binomial(2*j+n,j)*binomial(2*j+n+1,4*j+1)/(n+j+1),j,0,(n)/2); /* Vladimir Kruchinin, May 28 2014 */

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^2*(A+x*O(x^n))^4)); polcoeff(A, n)}

{a(n)=polcoeff((1/x)*serreverse( 2*x^3*(1+x)/(1 - sqrt(1-4*x^2*(1+x +x^3*O(x^n))^2))), 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^j*(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, (1-x*A^3)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A^3+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

x='x; y='y; Fxy = (1 + x*y)*(1 + x^2*y^4) - y;
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(27) \\ Gheorghe Coserea, Nov 30 2016

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(3*k) ).
(2) A(x) = (1/x)*Series_Reversion( 2*x^3*(1+x)/(1 - sqrt(1-4*x^2*(1+x)^2)) ).
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A104545 (Motzkin paths of length n having no consecutive (1,0) steps).
(4) A(x) = exp( Sum_{n>=1} x^n/n * (1-x*A(x)^3)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2 * x^k * A(x)^(3*k) ).
a(n) = sum(j=0..n/2, binomial(2*j+n,j)*binomial(2*j+n+1,4*j+1)/(n+j+1)). - Vladimir Kruchinin, May 28 2014
a(n) ~ sqrt((1 + 2*r*s^3 + 3*r^2*s^4)/(2*Pi*s*(3 + 5*r*s))) / (2*n^(3/2)*r^(n+1/2)), where r = 0.187614989725738719..., s = 1.61178302212918247... are roots of the system of equations r + 4*r^2*s^3 + 5*r^3*s^4 = 1, (1+r*s)*(1+r^2*s^4) = s. - Vaclav Kotesovec, May 28 2014

A215654 G.f. satisfies: A(x) = (1 + xA(x)^2) (1 + x*A(x)^3).

Original entry on oeis.org

1, 2, 11, 81, 684, 6257, 60325, 603641, 6210059, 65272503, 697898849, 7566847547, 82999675563, 919376968734, 10269588489433, 115548651723889, 1308374198000780, 14897993185500455, 170482798370871370, 1959574731164246402, 22614008012647634411, 261915716386286916342

Offset: 0

Paul D. Hanna, Aug 19 2012

nonn

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = (1 + x^r*F(x)^(p+1)) * (1 + x^(r+s)*F(x)^(p+q+1)), then
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ).
The radius of convergence of g.f. A(x) is r = 0.08035832347291483065438962031... with A(r) = 1.5393913914574609282262181402132760790902539070... where y=A(r) satisfies 20*y^3 - 38*y^2 + 15*y - 6 = 0.
r = 1/(187/300*17^(2/3) + 119/75*17^(1/3) + 1273/300). - Vaclav Kotesovec, Sep 17 2013
Number of hybrid ternary trees with n internal nodes. [Hong and Park]. - N. J. A. Sloane, Mar 26 2014

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 81*x^3 + 684*x^4 + 6257*x^5 + 60325*x^6 +...
Related expansions.
A(x)^2 = 1 + 4*x + 26*x^2 + 206*x^3 + 1813*x^4 + 17032*x^5 +...
A(x)^3 = 1 + 6*x + 45*x^2 + 383*x^3 + 3519*x^4 + 34023*x^5 +...
A(x)^5 = 1 + 10*x + 95*x^2 + 925*x^3 + 9270*x^4 + 95237*x^5 +...
where A(x) = 1 + x*(A(x)^2 + A(x)^3) + x^2*A(x)^5.
The g.f. also satisfies the series:
A(x) = 1 + 2*x*A(x)^2 + 3*x^2*A(x)^4 + 5*x^3*A(x)^6 + 8*x^4*A(x)^8 + 13*x^5*A(x)^10 + 21*x^6*A(x)^12 + 34*x^7*A(x)^14 +...+ Fibonacci(n+2)*x^n*A(x)^(2*n) +...
and consequently, A( x*(1-x-x^2)^2/(1+x)^2 ) = (1+x)/(1-x-x^2).
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x))*x*A(x) + (1 + 2^2*A(x) + A(x)^2)*x^2*A(x)^2/2 +
(1 + 3^2*A(x) + 3^2*A(x)^2 + A(x)^3)*x^3*A(x)^3/3 +
(1 + 4^2*A(x) + 6^2*A(x)^2 + 4^2*A(x)^3 + A(x)^4)*x^4*A(x)^4/4 +
(1 + 5^2*A(x) + 10^2*A(x)^2 + 10^2*A(x)^3 + 5^2*A(x)^4 + A(x)^5)*x^5*A(x)^5/5 +...
Explicitly,
log(A(x)) = 2*x + 18*x^2/2 + 185*x^3/3 + 2006*x^4/4 + 22412*x^5/5 + 255249*x^6/6 + 2946155*x^7/7 + 34342270*x^8/8 +...+ L(n)*x^n/n +...
where L(n) = [x^n] (1+x)^(2*n)/(1-x-x^2)^(2*n) / 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235. See p. 233. - _N. J. A. Sloane_, Mar 26 2014
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Cf. A007863, A198953, A215624, A198951.

Column k=3 of A245049.

a:= n-> coeff(series(RootOf((1+x*A^2)*(1+x*A^3)-A, A), x, n+1), x, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Apr 04 2019

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

a(n):=sum(binomial(2*n+i,i)*binomial(2*n+i+1,n-i),i,0,n)/(2*n+1); /* Vladimir Kruchinin, Apr 04 2019 */

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1 + x*A^2)*(1 + x*A^3)); polcoeff(A, n)}

{a(n)=polcoeff(sqrt((1/x)*serreverse( x*(1-x-x^2)^2/(1+x +x*O(x^n))^2)), n)}
for(n=0,31,print1(a(n),", "))

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

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

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

G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x-x^2)^2/(1+x)^2 ) ).
(2) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^k ).
(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(2*n)/n * Sum_{k=0..n} C(n,k)^2 / A(x)^k ).
(4) A(x) = Sum_{n>=0} Fibonacci(n+2) * x^n * A(x)^(2*n).
(5) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A007863 (number of hybrid binary trees with n internal nodes).
The formal inverse of g.f. A(x) is (sqrt(1-2*x+5*x^2) - (1+x))/(2*x^3).
a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(2*n+1) / (2*n+1).
Recurrence: 100*(n-1)*n*(2*n-1)*(2*n+1)*(4913*n^3 - 26877*n^2 + 49912*n - 30480)*a(n) = 2*(n-1)*(2*n-1)*(6254249*n^5 - 40468670*n^4 + 99110119*n^3 - 109861414*n^2 + 52822608*n - 8566560)*a(n-1) - 3*(2343501*n^7 - 22194333*n^6 + 87905623*n^5 - 187987155*n^4 + 233161624*n^3 - 166253172*n^2 + 62010112*n - 8952000)*a(n-2) + 6*(n-2)*(2*n-5)*(3*n-8)*(3*n-4)*(4913*n^3 - 12138*n^2 + 10897*n - 2532)*a(n-3). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ 1/1020*sqrt(73695 + 11730*17^(2/3) + 28815*17^(1/3)) * (187/300*17^(2/3) + 119/75*17^(1/3) + 1273/300)^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013
a(n) = 1/(2*n+1)*Sum_{i=0..n} C(2*n+i,i)*C(2*n+i+1,n-i). - Vladimir Kruchinin, Apr 04 2019

A007788 Number of augmented Andre 3-signed permutations: E.g.f. (1-sin(3*x))^(-1/3).

Original entry on oeis.org

1, 1, 4, 19, 136, 1201, 13024, 165619, 2425216, 40132801, 740882944, 15091932019, 336257744896, 8134269015601, 212309523595264, 5946914908771219, 177934946000306176, 5663754614516217601, 191097349696090537984, 6812679868133940475219, 255885704427935576621056

Offset: 0

R. Ehrenborg (ehrenbor(AT)lacim.uqam.ca) and M. A. Readdy (readdy(AT)lacim.uqam.ca)

nonn

It appears that all members are of the form 3k+1. - Ralf Stephan, Nov 12 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..200
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Preprint submitted to Ann. Sci. Math. Quebec, 1994. (Annotated scanned copy)
R. Ehrenborg and M. A. Readdy, Sheffer posets and r-signed permutations, Ann. Sci. Math. Québec, 19 (1995), no. 2, 173-196.
R. Ehrenborg and M. A. Readdy, The r-cubical lattice and a generalization of the cd-index, European J. Combin. 17 (1996), no. 8, 709-725.

Cf. A001586, A144015, A230134, A227544, A235128, A230114.
Cf. A007863, A235135, A235132.
Cf. A007559, A136630.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( (1-Sin(3*x))^(-1/3) ))); // G. C. Greubel, Mar 05 2020

m:=20; S:=series( (1-sin(3*x))^(-1/3), x, m+1): seq(j!*coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 05 2020

With[{nn=20},CoefficientList[Series[(1-Sin[3x])^(-1/3),{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Nov 23 2011 *)

Vec(serlaplace( (1-sin(3*x))^(-1/3) +O('x^20) )) \\ G. C. Greubel, Mar 05 2020

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k)*(3*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

m=20;
def A007788_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1-sin(3*x))^(-1/3) ).list()
a=A007788_list(m+1); [factorial(n)*a[n] for n in (0..m)] # G. C. Greubel, Mar 05 2020

E.g.f.: (1-sin(3*x))^(-1/3).
a(n) ~ n! * 2*6^n/(Pi^(n+2/3)*n^(1/3)*Gamma(2/3)). - Vaclav Kotesovec, Jun 25 2013
a(n) = Sum_{k=0..n} A007559(k) * (3*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A245049 Number A(n,k) of hybrid k-ary trees with n internal nodes; square array A(n,k), n>=0, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 7, 5, 1, 2, 11, 31, 8, 1, 2, 15, 81, 154, 13, 1, 2, 19, 155, 684, 820, 21, 1, 2, 23, 253, 1854, 6257, 4575, 34, 1, 2, 27, 375, 3920, 24124, 60325, 26398, 55, 1, 2, 31, 521, 7138, 66221, 331575, 603641, 156233, 89, 1, 2, 35, 691, 11764, 148348, 1183077, 4736345, 6210059, 943174, 144

Offset: 0

Alois P. Heinz, Jul 10 2014

			Square array A(n,k) begins:
   1,    1,     1,      1,       1,       1,       1, ...
   2,    2,     2,      2,       2,       2,       2, ...
   3,    7,    11,     15,      19,      23,      27, ...
   5,   31,    81,    155,     253,     375,     521, ...
   8,  154,   684,   1854,    3920,    7138,   11764, ...
  13,  820,  6257,  24124,   66221,  148348,  290305, ...
  21, 4575, 60325, 331575, 1183077, 3262975, 7585749, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235

Columns k=1-10 give: A000045(n+2), A007863, A215654, A239107, A239108, A239109, A245050, A245051, A245052, A245053.
Rows n=0-2 give: A000012, A007395, A004767(k-1).
Main diagonal gives A245054.

A:= (n, k)-> add(binomial((k-1)*n+i, i)*
    binomial((k-1)*n+i+1, n-i), i=0..n)/((k-1)*n+1):
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);

A[n_, k_] := Sum[Binomial[(k-1)*n+i, i]*Binomial[(k-1)*n+i+1, n-i], {i, 0, n}]/((k-1)*n+1); Table[A[n, 1+d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 18 2017, translated from Maple *)

A(n,k) = 1/((k-1)*n+1) * Sum_{i=0..n} C((k-1)*n+i,i)*C((k-1)*n+i+1,n-i).
A(n,k) = [x^n] ((1+x)/(1-x-x^2))^((k-1)*n+1) / ((k-1)*n+1).
G.f. for column k satisfies: A_k(x) = (1+x*A_k(x)^(k-1)) * (1+x*A_k(x)^k).

A011270 Hybrid binary rooted trees with n nodes whose root is labeled by "n".

Original entry on oeis.org

1, 1, 4, 18, 90, 481, 2690, 15547, 92124, 556664, 3417062, 21248966, 133576724, 847465593, 5419399722, 34895368578, 226050057378, 1472170887755, 9633297762870, 63305402213336, 417612181048826, 2764492667188504, 18358282050480384, 122265756020847943

Offset: 0

pallo(AT)u-bourgogne.fr (Jean Pallo)

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 18*x^3 + 90*x^4 + 481*x^5 + 2690*x^6 + 15547*x^7 + 92124*x^8 + 556664*x^9 + 3417062*x^10 + ...
where x = x*A(x) - x^2*A(x)^2/(1 - x*A(x))^2.

Alois P. Heinz, Table of n, a(n) for n = 0..500
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
J. M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50, 1994, 135-145.
Index entries for reversions of series
Index entries for sequences related to rooted trees

Cf. A011272.

G:= proc(n) option remember; if n<=0 then 1 else convert(series(
      (x^2*G(n-1)^3 +x*G(n-1)^2 +1)/ (1-x), x=0, n+1), polynom) fi
    end:
a:= n-> coeff(1+x*G(n-1)^2, x, n):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 22 2008
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 1, 4][n+1], (
      6*n*(210*n^2-411*n+163)*a(n-1)-4*(n-2)*(7*n-6)*(5*n-3)*a(n-2)
      +2*(n-3)*(2*n-3)*(35*n-16)*a(n-3))/(5*n*(n+1)*(35*n-51)))
    end:
seq(a(n), n=0..25); # Alois P. Heinz, May 18 2013

a[0] = 1; a[n_] := n*HypergeometricPFQ[{1-n, n+1, n+2}, {3/2, 2}, -1/4]; Table[ a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 02 2015, after Vladimir Kruchinin *)

G.f.: = 1+x*G(x)^2, where G(x) is g.f. for A007863.
Reversion of x - (x/(1 - x))^2 = 0, 1, -1, -2, -3, -4, -5, ... - Olivier Gérard, Jul 05 2001
a(n) = (2/(n+2))*Sum_{j=0...n} binomial(n+j+1, n+1)*binomial(n+j+2, n-j). - Vladimir Kruchinin, Dec 24 2010
G.f. A(x) satisfies: A(x) = 1/(1 - Sum_{k>=1} k*x^k*A(x)^k). - Ilya Gutkovskiy, Apr 10 2018
G.f. A(x) satisfies: A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n*A(x)^(n+1) / (1 + (n-1)*x*A(x))^(n+1). - Paul D. Hanna, Oct 08 2023
a(n) ~ sqrt((35 + (869750 - 5250*sqrt(105))^(1/3) + 5*(14*(497 + 3*sqrt(105)))^(1/3))/525) / (sqrt(Pi) * n^(3/2) * ((2 - 104/(-181 + 105*sqrt(105))^(1/3) + (-181 + 105*sqrt(105))^(1/3))/6)^n). - Vaclav Kotesovec, Oct 08 2023

A364374 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^2).

Original entry on oeis.org

1, 0, -1, 1, 2, -6, -1, 28, -31, -98, 288, 131, -1730, 1638, 7431, -19583, -15502, 135642, -99523, -664050, 1535896, 1816196, -11902728, 5944326, 64487669, -129346490, -213116764, 1112382523, -277762230, -6572175490, 11287106695, 25078981772, -107983368519, -1826241850

Offset: 0

Seiichi Manyama, Jul 21 2023

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

Cf. A364375, A364376.

Cf. A057078, A007863, A364371.

A364374 := proc(n)
    add( (-1)^k*binomial(n+k+1,k) * binomial(n+k+1,n-k)/(n+k+1),k=0..n) ;
end proc:
seq(A364374(n),n=0..80); # R. J. Mathar, Jul 25 2023

nmax = 33;
A[_] = 1;
Do[A[x_] = (1+x*A[x])*(1-x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 21 2023 *)

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k+1,k) * binomial(n+k+1,n-k) / (n+k+1).
D-finite with recurrence 15*n*(n+1)*a(n) +2*n*(13*n-11)*a(n-1) +12*(9*n^2-19*n+9)*a(n-2) +2*(10*n^2-65*n+99)*a(n-3) -4*(n-3)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Jul 25 2023
A(x) = (1/x) * series_reversion(x*(1 + x + x^2)/(1 + x)). - Peter Bala, Sep 08 2024

cp's OEIS Frontend

A011365 Reciprocal of g.f. for A007863.

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

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A198951 G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^3*A(x)^3).

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A192415 G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^3*A(x)^2).

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A198957 G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^2*A(x)^4).

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A215654 G.f. satisfies: A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^3).

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

A198951 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^3*A(x)^3).

A192415 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^3*A(x)^2).

A198957 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^2*A(x)^4).

A215654 G.f. satisfies: A(x) = (1 + xA(x)^2) (1 + x*A(x)^3).

A364374 G.f. satisfies A(x) = (1 + xA(x)) (1 - x*A(x)^2).