A069271 -id:A069271 - cp's OEIS frontend

A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

1, 1, 1, 1, 2, 4, 1, 3, 9, 22, 1, 4, 15, 52, 140, 1, 5, 22, 91, 340, 969, 1, 6, 30, 140, 612, 2394, 7084, 1, 7, 39, 200, 969, 4389, 17710, 53820, 1, 8, 49, 272, 1425, 7084, 32890, 135720, 420732, 1, 9, 60, 357, 1995, 10626, 53820, 254475, 1068012, 3362260

Offset: 0

Henry Bottomley, Mar 12 2002

For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).

Antidiagonals of convolution matrix of Table 1.5, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

			Rows start
  1;
  1,   1;
  1,   2,   4;
  1,   3,   9,  22;
  1,   4,  15,  52, 140;
etc.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

Columns include A000012, A000027, A055999.
Right-hand diagonals include A002293, A069271, A006632.
Cf. A130458 (row sums).

A069270 := proc(n,k)
        binomial(n+3*k,k)*(n-k+1)/(n+2*k+1) ;
end proc: # R. J. Mathar, Oct 11 2015

Table[Binomial[n + 3 k, k] (n - k + 1)/(n + 2 k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 27 2019 *)

T(n, k) = C(n+3k, k)*(n-k+1)/(n+2k+1).
For n >= k+3: T(n, k) = T(n-2, k+1)-T(n-3, k+1).
T(n, n) = T(n+2, n-1) = C(4n, n)/(3n+1).

A333094 a(n) is the n-th order Taylor polynomial (centered at 0) of c(x)^(2n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4x))/(2*x) is the o.g.f. of the Catalan numbers A000108.

Original entry on oeis.org

1, 3, 19, 144, 1171, 9878, 85216, 746371, 6609043, 59008563, 530279894, 4790262348, 43458522976, 395683988547, 3613641184739, 33088666355144, 303670285138067, 2792497004892302, 25724693177503987, 237350917999324431, 2193027397174233046, 20288470364637624223

Offset: 0

Peter Bala, Mar 15 2020

The sequence satisfies the Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k.
We conjecture that the sequence satisfies the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. Examples of these congruences are given below.
More generally, for each integer m, we conjecture that the sequence a_m(n) defined as the n-th order Taylor polynomial of c(x)^(m*n) evaluated at x = 1 satisfies the same supercongruences. For cases A099837(m = -2), A100219 (m = -1), A000012 (m = 0), A333093 (m = 1), A333095 (m = 3), A333096 (m = 4), A333097 (m = 5).

			n-th order Taylor polynomial of c(x)^(2*n):
  n = 0: c(x)^0 = 1 + O(x)
  n = 1: c(x)^2 = 1 + 2*x + O(x^2)
  n = 2: c(x)^4 = 1 + 4*x + 14*x^2 + O(x^3)
  n = 3: c(x)^6 = 1 + 6*x + 27*x^2 + 110*x^3 + O(x^4)
  n = 4: c(x)^8 = 1 + 8*x + 44*x^2 + 208*x^3 + 910*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 2 = 3, a(2) = 1 + 4 + 14 = 19, a(3) = 1 + 6 + 27 + 110 = 144 and a(4) = 1 + 8 + 44 + 208 + 910 = 1171.
The triangle of coefficients of the n-th order Taylor polynomial of c(x)^(2*n), n >= 0, in descending powers of x begins
                                          row sums
  n = 0 |   1                                 1
  n = 1 |   2    1                            3
  n = 2 |  14    4    1                      19
  n = 3 | 110   27    6   1                 144
  n = 4 | 910  208   44   8   1            1171
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group.
Examples of supercongruences:
a(13) - a(1) = 395683988547 - 3 = (2^6)*(3^2)*(13^3)*312677 == 0 ( mod 13^3 ).
a(3*7) - a(3) = 20288470364637624223 - 144 = (7^3)*17*269*12934629208861 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 150194008594715226556753 - 9878 = (5^6)*2593*5471* 677584325533 == 0 ( mod 5^6 ).

Cf. A000108, A069271, A333090 through A333097.

seq(add(2*n/(2*n+k)*binomial(2*n+2*k-1, k), k = 0..n), n = 1..25);
#alternative program
c:= x -> (1/2)*(1-sqrt(1-4*x))/x:
G := (x, n) -> series(c(x)^(2*n), x, 76):
seq(add(coeff(G(x, n), x, n-k), k = 0..n), n = 0..25);

Table[SeriesCoefficient[((1 + x)^2 * (1 - x - Sqrt[(1 - 3*x)*(1 + x)]) / (2*x^2))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = Sum_{k = 0..n} 2*n/(2*n+k)*binomial(2*n+2*k-1, k) for n >= 1.
a(n) = [x^n] ( (1 + x)*c^2(x/(1 + x)) )^n.
O.g.f.: ( 1 + x*f'(x)/f(x) )/( 1 - x*f(x) ), where f(x) = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + ... = (1/x)*Revert( x/c^2(x) ) is the o.g.f. of A069271.
Row sums of the Riordan array ( 1 + x*f'(x)/f(x), x*f(x) ) belonging to the Hitting time subgroup of the Riordan group.
a(n) ~ 2^(8*n + 7/2) / (13 * sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Mar 28 2020
a(n) = Sum_{k = 0..n} n/(2*n+2*k)*binomial(2*n+2*k, k) for n >= 1. - Peter Bala, Apr 19 2024

A364333 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^6).

Original entry on oeis.org

1, 2, 17, 216, 3224, 52640, 910452, 16392140, 303996224, 5767278431, 111401778266, 2183535060362, 43319505976084, 868220464851417, 17552981176788200, 357544690982030744, 7330803752675100908, 151172599088871911072, 3133367418601958989295, 65242183918761533467216

Offset: 0

Seiichi Manyama, Jul 18 2023

nonn

Cf. A007863, A069271, A073157, A215654, A215715, A364331.
Cf. A239109, A364339, A364340.
Cf. A200718.

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

a(n) = Sum_{k=0..n} binomial(2*n+4*k+1,k) * binomial(2*n+4*k+1,n-k) / (2*n+4*k+1).

A370056 a(n) = 2(4n+1)!/(3*n+2)!.

Original entry on oeis.org

1, 2, 18, 312, 8160, 287280, 12751200, 684028800, 43062243840, 3113350732800, 254265345734400, 23153103246873600, 2326025084653670400, 255579097716214272000, 30491180727539051520000, 3925248256199788277760000, 542357159056633603178496000

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Cf. A365340, A370057, A370058.

Cf. A000142, A069271.

PARI
```
a(n) = 2*(4*n+1)!/(3*n+2)!;
```

E.g.f.: exp( 1/2 * Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A069271(n).
D-finite with recurrence 3*(3*n+2)*(3*n+1)*a(n) -8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Feb 22 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x)^(3/2))^2.
a(n) = 2 * Sum_{k=0..n} (3*n+2)^(k-1) * |Stirling1(n,k)|. (End)

A166135 Number of possible paths to each node that lies along the edge of a cut 4-nomial tree, that is rooted one unit from the cut.

Original entry on oeis.org

1, 1, 3, 7, 22, 65, 213, 693, 2352, 8034, 28014, 98505, 350548, 1256827, 4542395, 16517631, 60417708, 222087320, 820099720, 3040555978, 11314532376, 42243332130, 158196980682, 594075563613, 2236627194858, 8440468925400, 31921622746680, 120970706601255

Offset: 1

Rick Jarosh (rick(AT)jarosh.net), Oct 08 2009

nonn

This is the third member of an infinite series of infinite series, the first two being the Catalan and Motzkin integers. The Catalan numbers lie on the edge of cut 2-nomial trees, Motzkin integers on the edge of cut 3-nomial trees.
a(n) is the number of increasing unary-binary trees with associated permutation that avoids 213. For more information about increasing unary-binary trees with an associated permutation, see A245888. - Manda Riehl, Aug 07 2014
Number of positive walks with n steps {-2,-1,1,2} starting at the origin, ending at altitude 1, and staying strictly above the x-axis. - David Nguyen, Dec 16 2016

G. C. Greubel, Table of n, a(n) for n = 1..1000
Cyril Banderier, Christian Krattenthaler, Alan Krinik, Dmitry Kruchinin, Vladimir Kruchinin, David Tuan Nguyen, and Michael Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv:1609.06473 [math.CO], 2016.
Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
Jérémie Bettinelli, Éric Fusy, Cécile Mailler, and Lucas Randazzo, A bijective study of Basketball walks, arXiv:1611.01478 [math.CO], 2016.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Rick Jarosh, Illustration of 4-nomial graph The series is the one at the top.
Rick Jarosh, First 4096 terms of the series in a text file.
Rick Jarosh, Illustrates the sequence in context. The above reference gives the first 16 terms of the first 128 sequences in the family, of which this sequence is the third, the first being the Catalan numbers, the second the Motzkin integers, the fourth A104632.

A055113 is the third sequence from the top of the graph illustrated above.

Cf. A000108, A001006, A069271, A092765, A126120.

[(&+[Binomial(n,k)*Binomial(n,2*n-3*k-1): k in [0..n]])/n : n in [1..30]]; // G. C. Greubel, Dec 12 2019

seq( add(binomial(n,k)*binomial(n,2*n-3*k-1), k=0..n)/n, n=1..30); # G. C. Greubel, Dec 12 2019

Rest[CoefficientList[Series[(Sqrt[(2-2Sqrt[1-4x]-3x)/x]-1)/2, {x, 0, 30}],x]] (* Benedict W. J. Irwin, Sep 24 2016 *)

vector(30, n, sum(k=0,n, binomial(n,k)*binomial(n,2*n-3*k-1))/n ) \\ G. C. Greubel, Dec 12 2019

[sum(binomial(n,k)*binomial(n,2*n-3*k-1) for k in (0..n))/n for n in (1..30)] # G. C. Greubel, Dec 12 2019

a(n) = ((36*n+18)*A092765(n) + (11*n+9)*A092765(n+1))/(2*(5*n+3)*(2*n+3)) (based on guessed recurrence). - Mark van Hoeij, Jul 14 2010
A(x) satisfies A(x)+A(x)^2 = A000108(x)-1, a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1) * C(2*n,n-k)*C(2*k-2,k-1). - Vladimir Kruchinin, May 12 2012
G.f.: (sqrt((2 - 2*sqrt(1-4*x) - 3*x)/x) - 1)/2. - Benedict W. J. Irwin, Sep 24 2016
a(n) ~ 4^n/(sqrt(5*Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 25 2016
Conjecture: 2*n*(2*n+1)*a(n) + (17*n^2-53*n+24)*a(n-1) + 6*(-13*n^2+43*n-36)*a(n-2) - 108*(2*n-5)*(n-3)*a(n-3) = 0. - R. J. Mathar, Oct 08 2016
a(n) = (1/n)*Sum_{k=0..n} binomial(n,k)*binomial(n,2*n-3*k-1). - David Nguyen, Dec 31 2016
From Alexander Burstein, Dec 12 2019: (Start)
1 + x*A(x) = 1/C(-x*C(x)^2), where C(x) is the g.f. of A000108.
F(x) = x*(1+x*A(x)) = x/C(-x*C(x)^2) is a pseudo-involution, i.e., the series reversion of x*(1 + x*A(x)) is x*(1 - x*A(-x)).
The B-sequence of F(x) is A069271, i.e., F(x) = x + x*F(x)*A069271(x*F(x)). (End)

A381780 G.f. A(x) satisfies A(x) = (1 + xA(x)^2) C(x*A(x)^3), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 13, 122, 1348, 16317, 209366, 2797461, 38509302, 542367569, 7778173646, 113196865436, 1667497600735, 24816081138489, 372551391235504, 5635157636123317, 85797446797707896, 1313857342649814042, 20222887980813290849, 312694810135597988049, 4854881337618505385339

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A007852, A069271, A381772.

Cf. A000108.

a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(2*n+k+1, n-k)/(2*n+3*k+1));

a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(2*n+k+1,n-k)/(2*n+3*k+1).

A185113 Number of dissections of a convex (3n+3)-sided polygon into n pentagons and one triangle (up to equivalence).

Original entry on oeis.org

1, 3, 18, 130, 1020, 8379, 70840, 610740, 5340060, 47187580, 420412278, 3770221338, 33991902308, 307826695050, 2798052616800, 25514463687720, 233296537299228, 2138295980859588, 19639886707062280, 180724535020583400, 1665767679910654320, 15376467276901980315

Offset: 0

F. Chapoton, Feb 03 2011

nonn

This sequence counts dissections of a convex 3n+3-sided polygon into one triangle and n pentagons, modulo a simple equivalence relation. This equivalence relation is defined by moving the triangle according to a simple rule (not detailed here).
(The equivalence relation is not defined by a group, but by local moves. Consider the hexagon formed by a pentagon adjacent to the triangle. The local move is half-rotation of such hexagons.)
The terms seem to be odd exactly for indices in A002450. - F. Chapoton Mar 08 2020

			For n=0, there is just one triangle, so that a(0)=1. For n=1, one can dissect an hexagon in 6 ways into a pentagon and a triangle. In this case, the equivalence relation just relates every such dissection to its half rotated image, so that a(1)=3.

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

Cf. A001700, A002293, A174687, A069271, A000217, A000260.

f := RootOf(81*x - 8 - (75*x - 8)*Z + (288*x^2 - 30*x)*_Z^2 + (256*x^3 - 27*x^2)*_Z^4): seq(coeff(series(f, x, 20), x, n), n = 0..19); # _Peter Luschny, Apr 06 2023

Table[Binomial[4*n + 1, n]*(n + 2)/(3*n + 2), {n, 0, 50}] (* G. C. Greubel, Jun 23 2017 *)

for(n=0,25, print1(binomial(4*n+1,n)*(n+2)/(3*n+2), ", ")) \\ G. C. Greubel, Jun 23 2017

def A185113(n):
    return binomial(4*n+1, n) * (n+2) / (3*n+2)

a(n) = binomial(4*n+1,n-1)*(n+2)/n = binomial(4*n+1,n)*(n+2)/(3*n+2).

a(n) = binomial(n+2,2) * A000260(n). - F. Chapoton Feb 22 2024

A268315 Decimal expansion of 256/27.

Original entry on oeis.org

9, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8, 1, 4, 8

Offset: 1

Gheorghe Coserea, Feb 01 2016

			9.481481481481481481481481481481...

Exponential growth rate for A000260, A002293, A005810, A006633, A006634, A052203, A066381, A069271, A078516, A078995, A153396, A196678, A268316.

[9] cat &cat[[4, 8, 1]^^45]; // Vincenzo Librandi, Feb 04 2016

Join[{9}, PadRight[{}, 120, {4, 8, 1}]] (* Vincenzo Librandi, Feb 04 2016 *)

PARI
```
1.0 * 256/27
```

More digits from Jon E. Schoenfield, Mar 15 2018

A379463 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -1.

Original entry on oeis.org

1, 1, 1, 1, 3, 11, 31, 71, 150, 334, 826, 2146, 5498, 13690, 33762, 84306, 214451, 551107, 1417291, 3637627, 9343555, 24096675, 62439587, 162331747, 422773098, 1102422546, 2879207046, 7534606366, 19756893196, 51894005428, 136496647696, 359478351816, 947912008073

Offset: 0

Emely Hanna Li Lobnig, Dec 23 2024

nonn

			For n = 4, the a(4)=3 paths are HHHH, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).

Cf. A069271, A127902, A379464.

A379463 := proc(n)
    add(2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2),k=0..floor(n/4)) ;
end proc:
seq(A379463(n),n=0..50) ; # R. J. Mathar, Jan 29 2025

a(n) = sum(k=0, floor(n/4), 2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2)) \\ Thomas Scheuerle, Jan 07 2025

a(n) ~ 2^(3/2) * (1 + 4/3^(3/4))^(n + 3/2) / (3^(11/8) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 15 2025
Conjecture D-finite with recurrence 3*n*(3*n+4)*(n-3)*(3*n+8)*a(n) +3*(-45*n^4+54*n^3+192*n^2-27*n-20)*a(n-1)
+9*(n-1)*(30*n^3-72*n^2-7*n+20)*a(n-2) -3*(n-1)*(n-2)*(90*n^2-234*n+95)*a(n-3) -(n-1)*(n-2)*(n-3)*(121*n+499)*a(n-4) +229*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 29 2025

More terms from Jinyuan Wang, Jan 07 2025

A179299 Number of corner-rooted pentagulations of girth 5 with 2n+1 inner faces.

Original entry on oeis.org

1, 5, 121, 4690, 228065, 12673173, 768897585, 49645423227, 3357669088200, 235393836387360, 16981887962145418, 1254065444086727685, 94424981678123285373, 7227272422780512414100, 560989900421822288646265, 44076648941211191411236261, 3500015582480750626266664105

Offset: 0

Jonathan Vos Post, Jul 09 2010

nonn

Olivier Bernardi and Éric Fusy, A bijection for triangulations, quadrangulations, pentagulations, etc, Journal of Combinatorial Theory, Series A 119, 1 (2012) 218-244; arXiv:1007.1292 [math.CO], 2010-2011.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
William G. Brown, Enumeration of quadrangular dissections of the disk, Canad. J. Math., 17 (1965) 302-317.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Cf. A069271, A001764, A179300.

k = 34;
{w0, w1, w2, w3} = FixedPoint[Function[{w0, w1, w2, w3}, {w1^2 + w2, w1^3 + 2 w1 w2 + w3, w1^4 + 3 w1^2 w2 + 2 w1 w3 + w2^2, x (1 + w0)^4} + O[x]^k] @@ # &, ConstantArray[0, 4]];
f = w3 - (w0 w3 + 2 w1 w2);
CoefficientList[f, x][[2 ;; ;; 2]]
(* Andrey Zabolotskiy, Jan 17 2022 *)

Entry edited, terms a(5) and beyond added by Andrey Zabolotskiy, Jan 17 2022

cp's OEIS Frontend

A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

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A333094 a(n) is the n-th order Taylor polynomial (centered at 0) of c(x)^(2*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108.

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A364333 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^6).

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A370056 a(n) = 2*(4*n+1)!/(3*n+2)!.

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A166135 Number of possible paths to each node that lies along the edge of a cut 4-nomial tree, that is rooted one unit from the cut.

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A381780 G.f. A(x) satisfies A(x) = (1 + x*A(x)^2) * C(x*A(x)^3), where C(x) is the g.f. of A000108.

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A185113 Number of dissections of a convex (3n+3)-sided polygon into n pentagons and one triangle (up to equivalence).

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Maple

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A268315 Decimal expansion of 256/27.

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A333094 a(n) is the n-th order Taylor polynomial (centered at 0) of c(x)^(2n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4x))/(2*x) is the o.g.f. of the Catalan numbers A000108.

A364333 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^6).

A370056 a(n) = 2(4n+1)!/(3*n+2)!.

A381780 G.f. A(x) satisfies A(x) = (1 + xA(x)^2) C(x*A(x)^3), where C(x) is the g.f. of A000108.