A060941 -id:A060941 - cp's OEIS frontend

A293946 a(n) = number of lattice paths from (0,0) to (3n,2n) which lie wholly below the line 3y=2x, only touching at the endpoints.

1, 2, 19, 293, 5452, 112227, 2460954, 56356938, 1332055265, 32251721089, 795815587214, 19939653287183, 505943824579282, 12974266405435153, 335717028959470883, 8754495459668971998, 229836484204401559180, 6069875377376291350173, 161145418968823760038557

Offset: 0

N. J. A. Sloane, Oct 24 2017

Robert Israel, Table of n, a(n) for n = 0..687 (corrected by Ray Chandler, Jan 19 2019)
M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.[Cached copy; Annotated copy of page 59]
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A000108, A060941, A322634.

f:= proc(n) local U,x,y;
  U:= Array(1..3*n,0..2*n);
  U[3*n,2*n]:= 1:
  for x from 3*n to 1 by -1 do
    for y from ceil(2/3*x)-1 to 0 by -1 do
      if x+1 <= 3*n then U[x,y]:= U[x+1,y] fi;
      if y+1 < 2/3*x or x=3*n then U[x,y]:= U[x,y]+U[x,y+1] fi;
  od od:
  U[1,0];
end proc:
map(f, [$1..30]); # Robert Israel, Oct 24 2017

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

a(n) = T(3n,2n) where T is the triangle from A294207. - Danny Rorabaugh, Oct 24 2017
G.f. A(z) satisfies A^10-19*A^9+162*A^8-816*A^7+2688*A^6+(-2*z-6048)*A^5+(19*z+9408)*A^4+(-73*z-9984)*A^3+(142*z+6912)*A^2+(-140*z-2816)*A+z^2+56*z+512=0 (Proven). - Bryan T. Ek, Oct 30 2017
a(n) ~ (2 + 10^(1/3)) * 5^(5*n - 3/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 1) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Sep 16 2021

More terms from Robert Israel, Oct 24 2017

Offset changed and a(0) by Danny Rorabaugh, Oct 24 2017

A300387 The number of paths of length 9n from the origin to the line y = 2x/7 with unit East and North steps that stay below the line or touch it.

Original entry on oeis.org

1, 4, 178, 11654, 900239, 76266406, 6853777795, 641688752961, 61916364799849, 6113859987916630, 614832988424140624, 62752222758863566993, 6483650829899569496380, 676834416167597357806799, 71278487569046416052210050, 7563527671079260544924794587, 807900192360879042402313084390

Offset: 0

Bryan T. Ek, Mar 04 2018

Equivalent to nonnegative walks from (0,0) to (9*n,0) with step set [1,2], [1,-7].

			For n=1, the possible walks are EEEEEEENN, EEEEEENEN, EEEEENEEN, EEEENEEEN.

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A001764, A060941, A300386, A300388, A300389, A274244.

terms = 17; f[_] = 0;
Do[f[t_] = f[t]^36 t^4 + 3 f[t]^29 t^3 - f[t]^28 t^3 + 4 f[t]^27 t^3 + 3 f[t]^22 t^2 - 2 f[t]^21 t^2 + 6 f[t]^20 t^2 - 3 f[t]^19 t^2 + 6 f[t]^18 t^2 + f[t]^15 t - f[t]^14 t + 2 f[t]^13 t - 2 f[t]^12 t + 3 f[t]^11 t - 3 f[t]^10 t + 4 f[t]^9 t + 1 + O[t]^terms, {terms}];
CoefficientList[f[t], t] (* Jean-François Alcover, Dec 04 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[Binomial[9*k, 2*k]*x^k/(9*k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2021 *)

G.f. satisfies: f=f^36*t^4+3*f^29*t^3-f^28*t^3+4*f^27*t^3+3*f^22*t^2-2*f^21*t^2+6*f^20*t^2-3*f^19*t^2+6*f^18*t^2+f^15*t-f^14*t+2*f^13*t-2*f^12*t+3*f^11*t-3*f^10*t+4*f^9*t+1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/9)*binomial(9*n, 2*n)*x^n/n ) - Bizley. Cf. A274244.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/9)*binomial(9*n-9*k, 2*n-2*k)*a(k) for n >= 1. (End)
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 4, 372, 39298, 4384884, 504464254, 59183637186, 7038517648243, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 11 (checked up to p = 101). - Peter Bala, Sep 14 2021
a(n) ~ c * 3^(18*n) / (n^(3/2) * 2^(2*n) * 7^(7*n)), where c = 0.0389180896257538883301359279112039841187646397413254619045749515282872957... - Vaclav Kotesovec, Sep 16 2021

A381772 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 11, 83, 727, 6940, 70058, 735502, 7949031, 87851819, 988307647, 11279719247, 130286197186, 1520108988221, 17889102534329, 212095541328931, 2531001870925559, 30376237591559863, 366417240105654587, 4440000077166319993, 54020150448778625847, 659665548217188211288

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A060941, A381773, A381774, A381775.
Cf. A007852, A069271, A381780.
Cf. A000108, A274052.

my(N=30, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^2)/x)^(1/2))

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

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

A300388 The number of paths of length 11n from the origin to the line y = 2x/9 with unit East and North steps that stay below the line or touch it.

Original entry on oeis.org

1, 5, 345, 35246, 4255288, 563796161, 79264265868, 11612106079203, 1753402118587333, 270965910076404428, 42648418241303137766, 6813002989827352100145, 1101807202785456951146158, 180034116076502209139781574, 29677341363243548521326632028, 4929368173228370040701922315332

Offset: 0

Bryan T. Ek, Mar 04 2018

Equivalent to nonnegative walks from (0,0) to (11*n,0) with step set [1,2], [1,-9].

			For n=1, the walks are EEEEEEEEENN, EEEEEEEENEN, EEEEEEENEEN, EEEEEENEEEN, EEEEENEEEEN.

M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.

Cf. A001764, A060941, A300386, A300387, A300389, A274256.

terms = 16; f[_] = 0;
Do[f[t_] = f[t]^55 t^5 + 4 f[t]^46 t^4 - f[t]^45 t^4 + 5 f[t]^44 t^4 + 6 f[t]^37 t^3 - 3 f[t]^36 t^3 + 12 f[t]^35 t^3 - 4 f[t]^34 t^3 + 10 f[t]^33 t^3 + 4 f[t]^28 t^2 - 3 f[t]^27 t^2 + 9 f[t]^26 t^2 - 6 f[t]^25 t^2 + 12 f[t]^24 t^2 - 6 f[t]^23 t^2 + 10 f[t]^22 t^2 + f[t]^19 t - f[t]^18 t + 2 f[t]^17 t - 2 f[t]^16 t + 3 f[t]^15 t - 3 f[t]^14 t + 4 f[t]^13 t - 4 f[t]^12 t + 5 f[t]^11 t + 1 + O[t]^terms, {terms}];
CoefficientList[f[t], t] (* Jean-François Alcover, Dec 04 2018 *)
nmax = 20; CoefficientList[Series[Exp[Sum[Binomial[11*k, 2*k]*x^k/(11*k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 16 2021 *)

G.f. satisfies: f = f^55*t^5 + 4*f^46*t^4 - f^45*t^4 + 5*f^44*t^4 + 6*f^37*t^3 - 3*f^36*t^3 + 12*f^35*t^3 - 4*f^34*t^3 + 10*f^33*t^3 + 4*f^28*t^2 - 3*f^27*t^2 + 9*f^26*t^2 - 6*f^25*t^2 + 12*f^24*t^2 - 6*f^23*t^2 + 10*f^22*t^2 + f^19*t - f^18*t + 2*f^17*t - 2*f^16*t + 3*f^15*t - 3*f^14*t + 4*f^13*t - 4*f^12*t + 5*f^11*t + 1.
From Peter Bala, Jan 03 2019: (Start)
O.g.f.: A(x) = exp( Sum_{n >= 1} (1/11)*binomial(11*n, 2*n)*x^n/n ) - Bizley. Cf. A274256.
Recurrence: a(0) = 1 and a(n) = (1/n) * Sum_{k = 0..n-1} (1/11)*binomial(11*n-11*k, 2*n-2*k)*a(k) for n >= 1. (End)
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 5, 715, 116213, 19954187, 3532860880, 637870220023, 116749388814357, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 5 except for p = 11 (checked up to p = 101). - Peter Bala, Sep 14 2021
a(n) ~ c * 11^(11*n) / (n^(3/2) * 2^(2*n) * 3^(18*n)), where c = 0.0304820662333129164912550234496338371466905844787974500412037592866845093... - Vaclav Kotesovec, Sep 16 2021

A381774 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^4 ) )^(1/4), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 19, 255, 3995, 68344, 1237526, 23316295, 452385355, 8977539540, 181374792040, 3718002102747, 77138798530854, 1616741658725930, 34179703551312530, 728019711835819493, 15608122038151106507, 336551042553481867640, 7293934071668996347055

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A060941, A381772, A381773, A381775.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^4)/x)^(1/4))

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

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

A381753 Expansion of exp( Sum_{k>=1} binomial(5k-1,2k-1) * x^k/k ).

Original entry on oeis.org

1, 4, 50, 846, 16495, 349240, 7803823, 181135830, 4324897697, 105543188190, 2620784850325, 66005699547352, 1682046970846570, 43291586055360034, 1123707191010320955, 29382536610737191930, 773229801368332554273, 20463493681189771623960

Offset: 0

Seiichi Manyama, Mar 06 2025

Cf. A006013, A079489, A182960, A381751, A381752, A381757, A381758.

Cf. A060941.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, binomial(5*k-1, 2*k-1)*x^k/k)))

a(n) = 2*sum(k=0, n, binomial(5*n+2*k+2, k)*binomial(5*n+2, n-k)/(5*n+2*k+2));

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(5*k-1,2*k-1) * a(n-k).
G.f.: B(x)^2, where B(x) is the g.f. of A060941.
a(n) = 2 * Sum_{k=0..n} binomial(5*n+2*k+2,k) * binomial(5*n+2,n-k)/(5*n+2*k+2).

A381775 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^6 ) )^(1/6), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 27, 523, 11871, 294668, 7747698, 212054604, 5978347887, 172421233231, 5063192676597, 150872475295522, 4550458484780442, 138652322209300991, 4261638256558924407, 131973650298641750844, 4113788296015093994719, 128973000885015536107140

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A060941, A381772, A381773, A381774.

Cf. A000108.

my(N=20, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^6)/x)^(1/6))

G.f. A(x) satisfies A(x) = (1 + x*A(x)^6) * C(x*A(x)^6).
a(n) = Sum_{k=0..n} binomial(6*n+2*k+1,k) * binomial(6*n+1,n-k)/(6*n+2*k+1).
a(n) = binomial(1 + 6*n, n)*hypergeom([-n, 1/2+3*n, 1+3*n], [2+5*n, 2+6*n], -4)/(1 + 6*n). - Stefano Spezia, Mar 07 2025

A322634 Sum of attendance numbers of all histories of length 5*n in the Bizley-Duchon's club model, divided by 5.

Original entry on oeis.org

5, 153, 4537, 133189, 3891675, 113415423, 3299905647

Offset: 1

Hugo Pfoertner, Dec 22 2018

The Bizley-Duchon's club model is equivalent to the lattice paths from (0,0) to (3*n,2*n) described in A293946. The attendance history of the club consists of persons entering in pairs and leaving in groups of three. The club closes when no persons are remaining. a(k)/A293946(k) is proportional to the mean area under the "filling level curve" of the club.

Banderier et al. show that the mean area is asymptotic to K*n^(3/2), with K=(1/2)*(15*Pi)^(1/2).

			a(1) = (15 + 10)/5 = 5:
  Contributions of the A293946(1) = 2 attendance histories are
  0 (+2) 2 (+2) 4 (+2) 6 (-3) 3 (-3) 0 -> 2 + 4 + 6 + 3 = 15
  0 (+2) 2 (+2) 4 (-3) 1 (+2) 3 (-3) 0 -> 2 + 4 + 1 + 3 = 10.

Cyril Banderier, Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area. Chassaing, Philippe and others. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AG, Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, pp.345-356, 2006, DMTCS Proceedings.
Cyril Banderier, Michael Wallner, Lattice paths of slope 2/5, arXiv:1605.02967 [cs.DM], 10 May 2016.

Cf. A060941, A293946.

A257995 Forests of binary shrubs on 3n vertices avoiding 321.

Original entry on oeis.org

1, 2, 37, 866, 23285, 679606, 20931998, 669688835, 22040134327, 741386199872, 25376258521393, 880977739374392, 30946637156662975, 1097929752363923490, 39284677690031136567, 1415992852373003788459

Offset: 0

Manda Riehl, May 15 2015

nonn

We define a shrub as a rooted, ordered tree with the only vertices being the root and leaves. We then label our shrubs' vertices with integers such that each child has a larger label than its parent. We associate a permutation to a tree by reading the labels from left to right by levels, starting with the root. A forest is an ordered collection of trees where all vertices in the forest have distinct labels. We associate a permutation to a forest by reading the permutation associated to each tree and then concatenating. We then enumerate labeled forests of binary shrubs whose associated permutation avoids 321.

David Bevan, Table of n, a(n) for n = 0..993
D Bevan, D Levin, P Nugent, J Pantone, L Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036, 2015
M. Riehl, Forests of binary shrubs avoiding patterns of length 3

A001764, A002293, A060941 and A144097 enumerate binary shrubs avoiding other patterns of length 3.

gf := RootOf(_Z^10*z^10+18*_Z^9*z^9+123*_Z^8*z^8+(-3*z^8+420*z^7+54*z^6)*_Z^7+(-36*z^7+751*z^6+486*z^5)*_Z^6+(-138*z^6+354*z^5+1053*z^4)*_Z^5+(3*z^6-228*z^5-213*z^4+162*z^3+729*z^2)*_Z^4+(18*z^5-215*z^4+2*z^3-360*z^2)*_Z^3+(15*z^4+24*z^3-71*z^2-54*z)*_Z^2+(-z^4+24*z^3-8*z^2+54*z-1)*_Z+4*z^2+4*z+1)^(1/2):
seq(coeff(series(gf,z,21),z,i),i=0..20);

b[k_]:=k(k+1)/2;n[k_]:=n[k]=Join[{b[k+1],b[k+1]-1},Table[b[i],{i,k,1,-1}],{1}];v[1]={1,0,1};v[k_]:=v[k]=Module[{s=MapIndexed[#1n[First@#2]&,v[k-1]]},Table[Total[If[i>Length@#,0,#[[i]]]&/@s],{i,Length@Last@s}]];a[k_]:=a[k]=Total@v[k];Array[a,20] (* David Bevan, Oct 27 2015 *)

A259550 a(n) = C(5n-1,2n)/3, n > 0, a(0) = 1.

Original entry on oeis.org

1, 2, 42, 1001, 25194, 653752, 17298645, 463991880, 12570420330, 343176898988, 9425842448792, 260170725132045, 7210477496434485, 200519284375732896, 5592628786362932776, 156375886125188595376, 4382048530314336892010, 123033460966787345446836

Offset: 0

Vladimir Kruchinin, Jun 30 2015

D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
V. V. Kruchinin and D. V. Kruchinin, Composita and its properties, J. Analysis and Number Theory 2 (2014), 1-8.
V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.
D. V. Kruchinin, On solving some functional equations, Advances in Difference Equations, Vol. 1 (2015), 1687-1847.

Cf. A000108, A167422, A001450, A060941.

[1] cat [Binomial(5*n-1, 2*n)/3: n in [1..20]]; // Vincenzo Librandi, Jul 01 2015

Join[{1}, Table[Binomial[5 n - 1, 2 n]/3, {n, 30}]] (* Vincenzo Librandi, Jul 01 2015 *)

makelist(if n=0 then 1 else binomial(5*n-1,2*n)/3,n,0,20);

vector(20, n, n--; if (n==0, 1, binomial(5*n-1,2*n)/3)) \\ Michel Marcus, Jul 01 2015

G.f.: A(x) = 1 + (x*B(x)')/(B(x)), B(x) = (1 + x*B(x)^5)*C(x*B(x)^5) is g.f. of A060941, C(x) is g.f. of Catalan numbers.
a(n) = n*Sum_{i = 0..n}((C(5*n,i)*C(7*n-2*i-1,n-i))/(6*n-i)), n > 1, a(0) = 1.
a(n) = 1/5*A001450(n) for n >= 1. exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 23*x^2 + 377*x^3 + ... is the o.g.f. for the sequence of Duchon numbers A060941. - Peter Bala, Oct 05 2015
D-finite with recurrence 6*n*(3*n-1)*(2*n-1)*(3*n-2)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

cp's OEIS Frontend

A293946 a(n) = number of lattice paths from (0,0) to (3n,2n) which lie wholly below the line 3y=2x, only touching at the endpoints.

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A300387 The number of paths of length 9*n from the origin to the line y = 2*x/7 with unit East and North steps that stay below the line or touch it.

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A381772 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

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A300388 The number of paths of length 11*n from the origin to the line y = 2*x/9 with unit East and North steps that stay below the line or touch it.

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A381774 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^4 ) )^(1/4), where C(x) is the g.f. of A000108.

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A381753 Expansion of exp( Sum_{k>=1} binomial(5*k-1,2*k-1) * x^k/k ).

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A381775 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^6 ) )^(1/6), where C(x) is the g.f. of A000108.

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A322634 Sum of attendance numbers of all histories of length 5*n in the Bizley-Duchon's club model, divided by 5.

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A257995 Forests of binary shrubs on 3n vertices avoiding 321.

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A300387 The number of paths of length 9n from the origin to the line y = 2x/7 with unit East and North steps that stay below the line or touch it.

A300388 The number of paths of length 11n from the origin to the line y = 2x/9 with unit East and North steps that stay below the line or touch it.

A381753 Expansion of exp( Sum_{k>=1} binomial(5k-1,2k-1) * x^k/k ).

A259550 a(n) = C(5n-1,2n)/3, n > 0, a(0) = 1.