A097188 -id:A097188 - cp's OEIS frontend

A025748 3rd-order Patalan numbers (generalization of Catalan numbers).

1, 1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620, 1324287183487521060

Offset: 0

Olivier Gérard

G.f. (with a(0)=0) is series reversion of x - 3*x^2 + 3*x^3.

The Hankel transform of a(n) is A005130(n) * 3^binomial(n,2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
I. M. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005, eq. (5.1).
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Apart from the initial 1, identical to A097188.

Cf. A004989, A005130, A034000, A034164, A073006,

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (4 - (1-9*x)^(1/3))/3 )); // G. C. Greubel, Sep 17 2019

A025748 :=proc(n)
        local x;
        coeftayl(4-(1-9*x)^(1/3),x=0,n) ;
        %/3 ;
end proc: # R. J. Mathar, Nov 01 2012

CoefficientList[Series[(4-Power[1-9x, (3)^-1])/3,{x,0,25}],x] (* Harvey P. Dale, Nov 14 2011 *)
Flatten[{1,Table[FullSimplify[9^(n-1) * Gamma[n-1/3] / (n * Gamma[2/3] * Gamma[n])],{n,1,25}]}] (* Vaclav Kotesovec, Feb 09 2014 *)
a[n_] := 9^(n-1) * Pochhammer[2/3, n-1]/n!; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n)=if(n<1,n==0,polcoeff(serreverse(x-3*x^2+3*x^3+x*O(x^n)),n))

def A025748_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((4 - (1-9*x)^(1/3))/3).list()
A025748_list(25) # G. C. Greubel, Sep 17 2019

From Wolfdieter Lang: (Start)
G.f.: (4 - (1-9*x)^(1/3))/3.
a(n) = 3^(n-1) * 2 * A034000(n-1)/n!, n >= 2.
a(n) = 3 * A034164(n-2), n >= 2. (End)
D-finite with recurrence n*a(n) + 3*(4-3*n)*a(n-1) = 0, n >= 2. - R. J. Mathar, Oct 29 2012
For n>0, a(n) = 9^(n-1) * Gamma(n-1/3) / (n * Gamma(2/3) * Gamma(n)). - Vaclav Kotesovec, Feb 09 2014
For n > 0, a(n) = 3^(2*n-1)*(-1)^(n+1)*binomial(1/3, n). - Peter Bala, Mar 01 2022
Sum_{n>=0} 1/a(n) = 37/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022
For n >= 1, a(n) = Integral_{x = 0..9} x^n * w(x) dx, where w(x) = 1/(2*sqrt(3)*Pi) *  x^(2/3)*(9 - x)^(1/3)/x^2. - Peter Bala, Oct 14 2024
a(n) ~ 9^(n-1) / (Gamma(2/3) * n^(4/3)). - Amiram Eldar, Aug 20 2025

A248324 Square array read by antidiagonals downwards: super Patalan numbers of order 3.

Original entry on oeis.org

1, 3, 6, 18, 9, 45, 126, 36, 45, 360, 945, 189, 135, 270, 2970, 7371, 1134, 567, 648, 1782, 24948, 58968, 7371, 2835, 2268, 3564, 12474, 212058, 480168, 50544, 15795, 9720, 10692, 21384, 90882, 1817640, 3961386, 360126, 94770, 47385, 40095, 56133, 136323, 681615, 15677145, 33011550, 2640924, 600210, 252720, 173745, 187110, 318087, 908820, 5225715, 135868590

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers of Gessel, A068555, based on Patalan numbers of order 3, A097188.

			T(0..4,0..4) is:
  1    3    18   126   945
  6    9    36   189   1134
  45   45   135  567   2835
  360  270  648  2268  9720
  2970 1782 3564 10692 40095

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A248325. First column is A004988, first row is A004987. a(n,1) = -A004990(n+1) = 3*A097188(n). a(1,k) = -A004989(k+1).

T(0,0)=1, T(n,k) = T(n-1,k)*(9*n-3)/(n+k), T(n,k) = T(n,k-1)*(9*k-6)/(n+k).

G.f.: (x/(1-9*x)^(2/3)+y/(1-9*y)^(1/3))/(x+y-9*x*y).

A369114 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^3) ).

Original entry on oeis.org

1, 3, 15, 92, 630, 4620, 35494, 282015, 2298417, 19108265, 161418543, 1381606044, 11955789440, 104427062460, 919430773992, 8151530382264, 72711166411422, 652075100808960, 5875868463764446, 53175058170610530, 483082193418731280, 4404057834071995110

Offset: 0

Seiichi Manyama, Jan 13 2024

nonn

Index entries for reversions of series

Cf. A368011, A369102.

Cf. A097188.

A369114 := proc(n)
    add(binomial(n+k,k) * binomial(4*n+2,n-3*k),k=0..floor(n/3)) ;
    %/(n+1) ;
end proc;
seq(A369114(n),n=0..70) ; # R. J. Mathar, Jan 25 2024

my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^3-x^3))/x)

a(n) = sum(k=0, n\3, binomial(n+k, k)*binomial(4*n+2, n-3*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} binomial(n+k,k) * binomial(4*n+2,n-3*k).

D-finite with recurrence 81*n*(n-1)*(n+1)*a(n) -945*n^2*(n-1)*a(n-1) +441*(n-1)*(3*n^2+9*n-20)*a(n-2) +3*(1039*n^3 -12393*n^2 +37406*n-33232)*a(n-3) -448*(2*n-5) *(4*n-13)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Jan 25 2024

A097186 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 3^n, where R_n(y) forms the initial (n+1) terms of g.f. A057083(y)^(n+1).

Original entry on oeis.org

1, 1, 6, 1, 9, 45, 1, 12, 78, 360, 1, 15, 120, 675, 2970, 1, 18, 171, 1134, 5859, 24948, 1, 21, 231, 1764, 10458, 51030, 212058, 1, 24, 300, 2592, 17334, 95256, 445824, 1817640, 1, 27, 378, 3645, 27135, 165726, 861597, 3905253, 15677145, 1, 30, 465, 4950, 40590, 272646, 1557765, 7760610, 34285680, 135868590

Offset: 0

Paul D. Hanna, Aug 03 2004

Row sums form A097189. Main diagonal is A004988. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A097188, where the g.f.s satisfy: A057083(x*A097188(x)) = A097188(x).

			Row polynomials evaluated at y=1/3 equals powers of 3:
3^1 = 1 + 6/3;
3^2 = 1 + 9/3 + 45/3^2;
3^3 = 1 + 12/3 + 78/3^2 + 360/3^3;
3^4 = 1 + 15/3 + 120/3^2 + 675/3^3 + 2970/3^4;
where A057083(y)^(n+1) has the same initial terms as the n-th row:
A057083(y) = 1 + 3y + 6y^2 + 9y^3 + 9y^4 + 0y^5 - 27y^6 +...
A057083(y)^2 = 1 + 6y +...
A057083(y)^3 = 1 + 9y + 45y^2 +...
A057083(y)^4 = 1 + 12y + 78y^2 + 360y^3 +...
A057083(y)^5 = 1 + 15y + 120y^2 + 675y^3 + 2970y^4 +...
Rows begin with n=0:
  1;
  1,  6;
  1,  9,  45;
  1, 12,  78,  360;
  1, 15, 120,  675,  2970;
  1, 18, 171, 1134,  5859,  24948;
  1, 21, 231, 1764, 10458,  51030, 212058;
  1, 24, 300, 2592, 17334,  95256, 445824, 1817640;
  1, 27, 378, 3645, 27135, 165726, 861597, 3905253, 15677145; ...

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Cf. A004988, A057083, A097188, A097189, A097190.

Table[SeriesCoefficient[3y/((1-9xy) - (1-3y)*(1-9xy)^(2/3)), {x,0,n}, {y,0,k}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2019 *)

{T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 3^n*(3^n -sum(j=0,n-1, T(n,j)/3^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)), x) +x*O(x^k))^((n+1)/n),k,x))))}

G.f.: A(x, y) = 3*y/((1-9*x*y) + (3*y-1)*(1-9*x*y)^(2/3)).

G.f.: A(x, y) = A004988(x*y)/(1 - x*A097188(x*y)).

More terms added by G. C. Greubel, Sep 17 2019

A369102 Expansion of (1/x) * Series_Reversion( x * ((1-x)^4-x^4) ).

Original entry on oeis.org

1, 4, 26, 204, 1772, 16408, 158752, 1585968, 16235472, 169423232, 1795611168, 19275231872, 209140483328, 2289981517312, 25271472702464, 280795784911616, 3138701648319744, 35270318924758016, 398215386792574464, 4515067063939210240, 51388662166213954560

Offset: 0

Seiichi Manyama, Jan 13 2024

nonn

Index entries for reversions of series

Cf. A097188, A151374.

Cf. A063021.

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

a(n) = sum(k=0, n\4, binomial(n+k, k)*binomial(5*n+3, n-4*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(n+k,k) * binomial(5*n+3,n-4*k).

A369124 Expansion of (1/x) * Series_Reversion( x * ((1-x)^4+x^4) ).

Original entry on oeis.org

1, 4, 26, 204, 1770, 16352, 157696, 1569096, 15988652, 165998624, 1749696208, 18673883696, 201394693864, 2191421381632, 24028822589440, 265238416143584, 2944999336948944, 32869042668479424, 368551132961138784, 4149643380825661824, 46897527236429235520

Offset: 0

Seiichi Manyama, Jan 13 2024

nonn

Index entries for reversions of series

Cf. A097188, A369123, A369125.

Cf. A369102.

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

a(n) = sum(k=0, n\4, (-1)^k*binomial(n+k, k)*binomial(5*n+3, n-4*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (-1)^k * binomial(n+k,k) * binomial(5*n+3,n-4*k).

A147630 a(1) = 1; for n>1, a(n) = Product_{k = 1..n-1} (9k - 3).

Original entry on oeis.org

1, 6, 90, 2160, 71280, 2993760, 152681760, 9160905600, 632102486400, 49303993939200, 4289447472710400, 411786957380198400, 43237630524920832000, 4929089879840974848000, 606278055220439906304000, 80028703289098067632128000, 11284047163762827536130048000

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 08 2008

Original name was: 9-factorial numbers (5).

Cf. A147629, A049211, A051232, A045756, A035012, A035013, A035017, A035018, A035020, A035022, A035023, A053116.

Cf. A048994, A097188, A132393.

[Round(9^n*Gamma(n+6/9)/Gamma(6/9)): n in [0..20]]; // Vincenzo Librandi, Feb 21 2015

s=1;lst={s};Do[s+=n*s;AppendTo[lst,s],{n,5,2*5!,9}];lst
Table[Product[9k-3,{k,1,n-1}],{n,20}] (* Harvey P. Dale, Sep 01 2016 *)

a(n):=n!*sum(binomial(k,n-k-1)*3^k*(-1)^(n-k-1)*binomial(n+k-1,n-1),k,1,n-1)/n; /* Vladimir Kruchinin, Apr 01 2011 */

a(n) = n--; prod(k=1, n, 9*k-3); \\ Michel Marcus, Feb 28 2015

a(n+1) = Sum_{k, 0<=k<=n}A132393(n,k)*6^k*9^(n-k). - Philippe Deléham, Nov 09 2008
a(n) = n!*(Sum_{k=1..n-1} binomial(k,n-k-1)*3^k*(-1)^(n-k-1)*binomial(n+k-1,n-1))/n for n>1, also a(n) = n!*A097188(n-1). - Vladimir Kruchinin, Apr 01 2011
a(n) = (-3)^n*sum_{k=0..n} 3^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = round(9^n * Gamma(n+6/9) / Gamma(6/9)). - Vincenzo Librandi, Feb 21 2015
Sum_{n>=1} 1/a(n) = 1 + (e/9^3)^(1/9)*(Gamma(2/3) - Gamma(2/3, 1/9)). - Amiram Eldar, Dec 21 2022

New name from Peter Bala, Feb 20 2015

More terms from Michel Marcus, Feb 28 2015

A369125 Expansion of (1/x) * Series_Reversion( x * ((1-x)^5+x^5) ).

Original entry on oeis.org

1, 5, 40, 385, 4095, 46375, 548300, 6689550, 83593250, 1064463125, 13762667750, 180189122750, 2384130651875, 31829162793750, 428227113655000, 5800188020157500, 79026653220693750, 1082367047392625000, 14893567523068062500, 205796463286063912500

Offset: 0

Seiichi Manyama, Jan 13 2024

nonn

Index entries for reversions of series

Cf. A097188, A369123, A369124.

Cf. A368011.

A369125 := proc(n)
    add((-1)^k * binomial(n+k,k) * binomial(6*n+4,n-5*k),k=0..floor(n/5)) ;
    %/(n+1) ;
end proc;
seq(A369125(n),n=0..70) ; # R. J. Mathar, Jan 25 2024

my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^5+x^5))/x)

a(n) = sum(k=0, n\5, (-1)^k*binomial(n+k, k)*binomial(6*n+4, n-5*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n+k,k) * binomial(6*n+4,n-5*k).

D-finite with recurrence 2*n*(n-1)*(n-2)*(24115*n-65551)*(n+1)*a(n) -5*n*(n-1) *(n-2)*(392783*n^2 -1296338*n +636787)*a(n-1) +100*(n-1)*(n-2) *(86231*n^3 -471376*n^2 +844569*n -522390)*a(n-2) +50*(n-2)*(2114435*n^4 -14778692*n^3 +35712085*n^2 -33505588*n +8727216)*a(n-3) +50*(13474985*n^5 -137009240*n^4 +513119690*n^3 -832716700*n^2 +478740305*n +26151216)*a(n-4) -125*(5*n-21) *(6943*n-12944) *(5*n-19)*(5*n-18)*(5*n-17)*a(n-5)=0. - R. J. Mathar, Jan 25 2024

A225439 Expansion of 3x/((1-(1-9x)^(1/3))(1-9x)^(2/3)).

Original entry on oeis.org

1, 3, 21, 162, 1305, 10773, 90342, 765936, 6546177, 56293380, 486451251, 4220183916, 36731240910, 320571837810, 2804298945840, 24580601689752, 215832643307217, 1898042178972285, 16714070686567620, 147360883148636850, 1300623629653125855

Offset: 0

Vladimir Kruchinin, May 08 2013

nonn

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

Cf. A025748, A097188, A122868.

A225439 := n -> `if`(n=0,1,(GAMMA(n+2/3)/GAMMA(2/3)+GAMMA(n+1/3)/(GAMMA(1/3)))* 3^(2*n-1)/GAMMA(n+1)): seq(A225439(i),i=0..20); # Peter Luschny, Jul 05 2013

Table[Sum[Binomial[k,n-k]*3^k*(-1)^(n-k)*Binomial[n+k-1,n-1], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, May 22 2013 *)

a(n):=if n=0 then 1 else sum(binomial(k,n-k)*3^(k)*(-1)^(n-k)*binomial(n+k-1,n-1),k,0,n);

my(x='x+O('x^66)); Vec(3*x/((1-(1-9*x)^(1/3))*(1-9*x)^(2/3))) \\ Joerg Arndt, May 08 2013

{a(n)=local(B=(1-(1-9*x+x^2*O(x^n))^(1/3))/(3*x));polcoeff(1+x*B'/B, n, x)} \\ Paul D. Hanna, May 08 2013

a(n) = Sum_{k = 0..n} C(k,n-k)*3^(k)*(-1)^(n-k)*C(n+k-1,n-1), n>0, a(0)=1.
G.f.: A(x) = 1 + x*B'(x)/B(x), where B(x) = (1-(1-9*x)^(1/3))/(3*x) is the g.f. of A097188.
n*(n-1)*a(n) = 18*(n-1)^2*a(n-1) - 9*(3*n-5)*(3*n-4)*a(n-2). - Vaclav Kotesovec, May 22 2013
a(n) ~ 3^(2*n-1)/(GAMMA(2/3)*n^(1/3)). - Vaclav Kotesovec, May 22 2013
a(n) = ((Gamma(n+2/3)/Gamma(2/3))+(Gamma(n+1/3)/Gamma(1/3)))*3^(2*n-1)/Gamma(n+1) for n > 0. - Peter Luschny, Jul 05 2013
From Peter Bala, Mar 11 2022: (Start)
a(n) = [x^n] (1/(1 - 3*x + 3*x^2))^n. Cf. A122868(n) = [x^n] (1 + 3*x + 3*x^2)^n.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)

A097189 Row sums of triangle A097186, in which the n-th row polynomial R_n(y) is formed from the initial (n+1) terms of g.f. A057083(y)^(n+1), where R_n(1/3) = 3^n for all n >= 0.

Original entry on oeis.org

1, 7, 55, 451, 3781, 32131, 275563, 2378971, 20640907, 179791327, 1571002291, 13762897435, 120832716655, 1062818450155, 9363143224315, 82600459304203, 729572125425661, 6450872644562491, 57092964352312951, 505729048454449651

Offset: 0

Paul D. Hanna, Aug 03 2004

nonn

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

Cf. A004988, A057083, A097186, A097188.

List([0..30], n-> 1 + Sum([0..n-1], k-> Sum([0..n-k], j-> (-1)^(n-k-j)*3^j*Binomial(j, n-k-j)*Binomial(n+j, n) )) ); # G. C. Greubel, Sep 17 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 3/((1-9*x) + 2*(1-9*x)^(2/3)) )); // G. C. Greubel, Sep 17 2019

seq(coeff(series(3/((1-9*x) + 2*(1-9*x)^(2/3)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[3/((1-9*x) + 2*(1-9*x)^(2/3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

a(n):=sum(sum(binomial(k,n-m-k)*3^k*(-1)^(n-m-k)*binomial(n+k,n),k,0,n-m),m,0,n-1)+1; /* Vladimir Kruchinin, Sep 09 2019 */

a(n)=polcoeff(3/((1-9*x)+2*(1-9*x+x*O(x^n))^(2/3)),n,x)

def A097189_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 3/((1-9*x) + 2*(1-9*x)^(2/3)) ).list()
A097189_list(30) # G. C. Greubel, Sep 17 2019

G.f.: A(x) = 3/((1-9*x) + 2*(1-9*x)^(2/3)).
G.f.: A(x) = A004988(x)/(1 - x*A097188(x)).
a(n) = 1 + Sum_{m=0..n-1} Sum_{k=0..n-m} C(k,n-m-k)*3^k*(-1)^(n-m-k)*C(n+k,n). - Vladimir Kruchinin, Sep 17 2019
Conjecture: n*(n-1)*a(n) - (19*n-18)*(n-1)*a(n-1) + 9*(11*n^2-31*n+22)*a(n-2) - 9*(3*n-4)*(3*n-5)*a(n-3) = 0. - R. J. Mathar, Nov 16 2012
a(n) ~ 3^(2*n+1)/(2*Gamma(2/3) * n^(1/3))*(1 - sqrt(3)*Gamma(2/3)^2 / (4*Pi*n^(1/3))). - Vaclav Kotesovec, Feb 04 2014

cp's OEIS Frontend

A025748 3rd-order Patalan numbers (generalization of Catalan numbers).

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A248324 Square array read by antidiagonals downwards: super Patalan numbers of order 3.

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A369114 Expansion of (1/x) * Series_Reversion( x * ((1-x)^3-x^3) ).

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A097186 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 3^n, where R_n(y) forms the initial (n+1) terms of g.f. A057083(y)^(n+1).

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A369102 Expansion of (1/x) * Series_Reversion( x * ((1-x)^4-x^4) ).

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A369124 Expansion of (1/x) * Series_Reversion( x * ((1-x)^4+x^4) ).

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A147630 a(1) = 1; for n>1, a(n) = Product_{k = 1..n-1} (9k - 3).

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PARI

Formula

A225439 Expansion of 3x/((1-(1-9x)^(1/3))(1-9x)^(2/3)).