A036765 -id:A036765 - cp's OEIS frontend

A064580 Triangle associated with rooted trees with a degree constraint (A036765).

1, 1, 1, 1, 2, 2, 1, 3, 5, 5, 1, 4, 9, 14, 13, 1, 5, 14, 28, 40, 36, 1, 6, 20, 48, 87, 118, 104, 1, 7, 27, 75, 161, 273, 357, 309, 1, 8, 35, 110, 270, 536, 866, 1100, 939, 1, 9, 44, 154, 423, 951, 1782, 2772, 3441, 2905, 1, 10, 54, 208, 630, 1572, 3310, 5928, 8946, 10900, 9118

Offset: 0

Henry Bottomley, Sep 21 2001

Main diagonal is A036765. - Paul D. Hanna, Nov 18 2016

			Triangle begins:
  1;
  1,  1;
  1,  2,  2;
  1,  3,  5,  5;
  1,  4,  9, 14, 13;
  1,  5, 14, 28, 40, 36;
  ...

Columns include A000012, A000027, A000096.
Main diagonal is A036765.
The sequence of triangles A010054 (triangle indicator), A007318 (Pascal), A026300 (Motzkin), A064580, ... converges to the triangle A009766 (Catalan).
Row sums give A159772.

a[n_, k_] /; 0 <= k <= n = a[n, k] = a[n - 1, k] + a[n - 1, k - 1] + a[n - 1, k - 2] + a[n - 1, k - 3]; a[0, 0] = 1; a[, ] = 0;
Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018 *)

# uses[riordan_array from A256893]
M = riordan_array(1, x/(1+x+x^2+x^3), 12).inverse()
for m in M[1:]:
    print([r for r in reversed(list(m)) if r != 0]) # Peter Luschny, Aug 17 2016

a(n, k) = a(n-1, k) + a(n-1, k-1) + a(n-1, k-2) + a(n-1, k-3) with a(0, 0)=1 and a(n, k)=0 if n < k or k < 0.

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 17 2007

A246555 a(n) = A036765(n-1) if n>0, with a(0) = 1.

Original entry on oeis.org

1, 1, 1, 2, 5, 13, 36, 104, 309, 939, 2905, 9118, 28964, 92940, 300808, 980864, 3219205, 10626023, 35252867, 117485454, 393133485, 1320357501, 4449298136, 15038769672, 50973266380, 173214422068, 589998043276, 2014026871496, 6889055189032, 23608722350440

Offset: 0

Michael Somos, Nov 14 2014

nonn

			G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 36*x^6 + 104*x^7 + 309*x^8 + ...

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

Cf. A036765, A210736.

{a(n) = if( n<1, n==0, polcoeff( serreverse( x / (1 + x + x^2 + x^3 + x*O(x^n))), n))};

Given g.f. A(x), series reversion of x * A(x) = x * A210736(-x).
G.f.: A(x) satisfies A(x) = 1 + x * A * (1 + (A(x) - 1)^2).
D-finite with recurrence 2*n*(2*n+1)*a(n) +3*(7*n^2-23*n+10) *a(n-1) +12*(-8*n^2+31*n-29) *a(n-2) -16*(5*n-11)*(n-3)*a(n-3) -128*(n-3) *(n-4)*a(n-4)=0. - R. J. Mathar, Jul 21 2023

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)

A243753 Number A(n,k) of Dyck paths of semilength n avoiding the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 2, 1, 4, 1, 1, 0, 0, 0, 1, 1, 2, 4, 1, 9, 1, 1, 0, 0, 0, 1, 1, 2, 4, 9, 1, 21, 1, 1, 0, 0, 0, 1, 1, 1, 4, 9, 21, 1, 51, 1, 1, 0, 0, 0

Offset: 0

Alois P. Heinz, Jun 09 2014

			Square array A(n,k) begins:
  1, 1, 1, 1, 1,   1, 1,   1,   1,    1, ...
  0, 0, 0, 1, 1,   1, 1,   1,   1,    1, ...
  0, 0, 0, 1, 1,   1, 1,   2,   2,    2, ...
  0, 0, 0, 1, 1,   2, 1,   4,   4,    4, ...
  0, 0, 0, 1, 1,   4, 1,   9,   9,    9, ...
  0, 0, 0, 1, 1,   9, 1,  21,  21,   23, ...
  0, 0, 0, 1, 1,  21, 1,  51,  51,   63, ...
  0, 0, 0, 1, 1,  51, 1, 127, 127,  178, ...
  0, 0, 0, 1, 1, 127, 1, 323, 323,  514, ...
  0, 0, 0, 1, 1, 323, 1, 835, 835, 1515, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns give: 0, 1, 2: A000007, 3, 4, 6: A000012, 5: A001006(n-1) for n>0, 7, 8, 14: A001006, 9: A135307, 10: A078481 for n>0, 11, 13: A105633(n-1) for n>0, 12: A082582, 15, 16: A036765, 19, 27: A114465, 20, 24, 26: A157003, 21: A247333, 25: A187256(n-1) for n>0.
Main diagonal gives A243754 or column k=0 of A243752.
Cf. A242450, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.

A:= proc(n, k) option remember; local b, m, r, h;
      if k<2 then return `if`(n=0, 1, 0) fi;
      m:= iquo(k, 2, 'r'); h:= 2^ilog2(k); b:=
      proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
        `if`(t=m and r=1, 0, b(x-1, y+1, irem(2*t+1, h)))+
        `if`(t=m and r=0, 0, b(x-1, y-1, irem(2*t, h)))))
      end; forget(b);
      b(2*n, 0, 0)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = Module[{b, m, r, h}, If[k<2, Return[If[n == 0, 1, 0]]]; {m, r} = QuotientRemainder[k, 2]; h = 2^Floor[Log[2, k]]; b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, If[t == m && r == 1, 0, b[x-1, y+1, Mod[2*t+1, h]]] + If[t == m && r == 0, 0, b[x-1, y-1, Mod[2*t, h]]]]]; b[2*n, 0, 0]]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

A059968 Number of 10-ary trees.

Original entry on oeis.org

1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, 4263421511271, 96723482198980, 2216905597676000, 51256802757808320, 1194060413809070710, 27999654303202465310, 660370070571422998410, 15654733143626084944150

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Mar 05 2001

From Wolfdieter Lang, Feb 06 2020: (Start)
Ninth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 10-Raney sequence.
a(n), n>=1, enumerates 10-ary trees (rooted, ordered, incomplete) with n vertices (including the root).
See Graham et al., Hilton and Pedersen, Hoggat and Bicknell, Frey and Sellers references given in A062993. (End)
This is instance k = 10 of the generalized Catalan family {C(k, n)}A130564%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564 - _Wolfdieter Lang, Feb 05 2024

			There are a(2)=10 10-ary trees (vertex degree <=10 and 10 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 10 trees yields 10*10+binomial(10,2)=145=a(3) such trees. - _Wolfdieter Lang_, Sep 14 2007.

G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.

S. Heubach, N. Y. Li and T. Mansour, Staircase tilings and k-Catalan structures, Discrete Math., 308 (2008), 5954-5964.

Related algebraic sequences concerning trees: strictly k-ary trees (A000108: s=x+s^2, A001263: s=(x, y)+(x, s)+(s, y)+(s, s))), (A001764: s=x+s^3), (A002293: s=x+s^4), (A002294: s=x+s^5), (A002295: s=x+s^6), (A002296: s=x+s^7), (A007556: s=x+s^8), at most k-ary trees (A001006: s=x+xs+xs^2), (A036765-A036769, s=x+xs^2....+xs^k, k=3, 4, 5, 6, 7).
Pfaff-Fuss A062993 column members: A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994, A230388.
Cf. A130564.

seq(binomial(10*k+1, k)/(9*k+1), k=0..30);
n:=30:G:=series(RootOf(g = 1+x*g^10, g), x=0, n+1):seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 01 2015

a[n_] := Binomial[10n, n]/(9n+1);
a /@ Range[0, 25] (* Jean-François Alcover, Jan 17 2020 *)

G.f. A(x) satisfies: A = x + A^10.
a(n) = binomial(k*n, n)/((k-1)*n+1), for k=10.
Recurrence: a(0) = 1; a(n) = Sum_{i1+i2+..i10=n-1} a(i1)*a(i2)*...*a(i10) for n>=1. - Robert FERREOL, Apr 01 2015
From Wolfdieter Lang, Feb 06 2020: (Start)
a(n) = A062993(n+8, 8). [Corrected by Robert FERREOL, Apr 01 2015]
G.f.: RootOf((_Z^10)*x-_Z+1) (Maple notation, from ECS, see links for A007556).
G.f.: hypergeometric([1, 2, 3, 4, 5, 6, 7, 8, 9]/10, [2, 3, 4, 5, 6, 7, 8, 10]/9, (10^10/9^9)*x),
E.g.f.: hypergeometric([1, 2, 3, 4, 5, 6, 7, 8, 9]/10, [2, 3, 4, 5, 6, 7, 8, 9, 10]/9, (10^10/9^9)*x).
For other family members see the crossreferences.
(End)
D-finite with recurrence 81*n*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n+1)*(3*n-2)*(9*n-4)*(9*n-2)*a(n) -800*(10*n-9)*(5*n-4)*(10*n-7)*(5*n-3)*(2*n-1)*(5*n-2)*(10*n-3)*(5*n-1)*(10*n-1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022
a(n) ~ (10^10/9^9)^n*sqrt(10/(2*Pi*(9*n)^3)). - Robert A. Russell, Jul 15 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^19). - Seiichi Manyama, Jun 16 2025

More terms from James Sellers, Mar 15 2001
a(0)=1 inserted by Alois P. Heinz, Jan 17 2020
A062744 merged into this sequence by Wolfdieter Lang, Feb 06 2020

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

A199874 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)^2).

Original entry on oeis.org

1, 1, 3, 10, 37, 147, 611, 2625, 11564, 51953, 237123, 1096420, 5125063, 24178427, 114974387, 550511901, 2651896733, 12843003108, 62494595022, 305400429548, 1498184696271, 7375179807191, 36421312544431, 180383163330765, 895756907248150, 4459095182031675, 22247684478181317

Offset: 0

Paul D. Hanna, Nov 11 2011

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 147*x^5 + 611*x^6 +...
where A( x/(1+x^2) - x^2 ) = (1+x^2)/(1-x-x^3).
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 26*x^3 + 103*x^4 + 428*x^5 + 1838*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 80*x^3 + 359*x^4 + 1632*x^5 + 7506*x^6 +...
where A(x) = 1 + x*(1+x)*A(x)^2 + x^3*A(x)^4.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x)*x*A(x) + (1 + 2^2*x + x^2)*x^2*A(x)^2/2 +
(1 + 3^2*x + 3^2*x^2 + x^3)*x^3*A(x)^3/3 +
(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^4*A(x)^4/4 +
(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)*x^5*A(x)^5/5 +
(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^6*A(x)^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 101*x^4/4 + 481*x^5/5 + 2330*x^6/6 +...

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

Cf. A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

Cf. A186241, A200074, A200075, A200718, A200719, A215576.

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^2)*(1+x^2*AGF^2)-AGF,x,j]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}];Flatten[{1,aa}] (* Vaclav Kotesovec, Aug 18 2013 *)

{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)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

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

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

G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x/(1+x^2) - x^2 ).
(2) A( x*(1-x-x^3)/(1+x^2) ) = (1+x^2)/(1-x-x^3).
(3) a(n) = [x^n] ((1+x^2)/(1-x-x^3))^(n+1) / (n+1).
(4) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k) * x^n*A(x)^n/n ).
(5) A(x) = exp( Sum_{n>=1} (1-x)^(2*n+1)*(Sum_{k>=0} C(n+k,k)^2*x^k) * x^n*A(x)^n/n ).
Recurrence: 31*(n-1)*n*(n+1)*(85396*n^4 - 902916*n^3 + 3471647*n^2 - 5767203*n + 3503250)*a(n) = 2*(n-1)*n*(6319304*n^5 - 69975436*n^4 + 290875210*n^3 - 559740413*n^2 + 484175751*n - 138985722)*a(n-1) + 2*(n-1)*(2903464*n^6 - 36506072*n^5 + 179801738*n^4 - 439606930*n^3 + 553204983*n^2 - 328951215*n + 67014378)*a(n-2) + 2*(2*n - 5)*(1964108*n^6 - 24695284*n^5 + 123902749*n^4 - 317652203*n^3 + 438313617*n^2 - 307740825*n + 85471038)*a(n-3) - 32*(n-3)*(2*n - 7)*(85396*n^5 - 860218*n^4 + 3249611*n^3 - 5747414*n^2 + 4753791*n - 1471338)*a(n-4) + 8*(n-4)*(n-3)*(2*n - 9)*(85396*n^4 - 561332*n^3 + 1275275*n^2 - 1191073*n + 390174)*a(n-5). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ c*d^n/n^(3/2), where d=5.28245622984... is the root of the equation -16 + 64*d - 92*d^2 - 68*d^3 - 148*d^4 + 31*d^5 = 0 and c = 0.49559010377906722118329... - Vaclav Kotesovec, Aug 18 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-2*k+1,k) * binomial(2*n-2*k+1,n-2*k) / (2*n-2*k+1). - Seiichi Manyama, Jul 18 2023

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

cp's OEIS Frontend

A064580 Triangle associated with rooted trees with a degree constraint (A036765).

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A246555 a(n) = A036765(n-1) if n>0, with a(0) = 1.

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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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A243753 Number A(n,k) of Dyck paths of semilength n avoiding the consecutive step pattern given by the binary expansion of k, where 1=U=(1,1) and 0=D=(1,-1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A059968 Number of 10-ary trees.

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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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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).

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

A199874 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)^2).

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).