A004988 -id:A004988 - cp's OEIS frontend

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).

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

A383601 Expansion of 1/( (1-x) * (1-10*x)^2 )^(1/3).

Original entry on oeis.org

1, 7, 58, 514, 4705, 43879, 414208, 3943492, 37782346, 363760390, 3515819020, 34088616940, 331383573010, 3228590970430, 31514912933800, 308126549765440, 3016908101224105, 29576113797737695, 290271761086278610, 2851684765215491050, 28040613734007656545

Offset: 0

Seiichi Manyama, May 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A383605, A383606.

Cf. A004988, A377233, A383597.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/( (1-x) * (1-10*x)^2 )^(1/3))); // Vincenzo Librandi, May 05 2025

Table[Sum[(-9)^k* Binomial[-2/3,k]* Binomial[n,k],{k,0,n}],{n,0,22}] (* Vincenzo Librandi, May 05 2025 *)

a(n) = sum(k=0, n, (-9)^k*binomial(-2/3, k)*binomial(n, k));

a(n) = Sum_{k=0..n} (-9)^k * binomial(-2/3,k) * binomial(n,k).
n*a(n) = (11*n-4)*a(n-1) - 10*(n-1)*a(n-2) for n > 1.
a(n) ~ Gamma(1/3) * 2^(n - 2/3) * 5^(n + 1/3) / (Pi * 3^(1/6) * n^(1/3)). - Vaclav Kotesovec, May 02 2025
a(n) = hypergeom([2/3, -n], [1], -9). - Stefano Spezia, May 04 2025

A216705 a(n) = Product_{k=1..n} (81 - 9/k).

Original entry on oeis.org

1, 72, 5508, 429624, 33832890, 2679564888, 213025408596, 16981168285224, 1356370816782267, 108509665342581360, 8691624193940766936, 696910230823250585232, 55927046023565859464868, 4491372003738673637024784, 360913821729000560118063000

Offset: 0

Michel Lagneau, Sep 16 2012

nonn

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Harvey P. Dale, Table of n, a(n) for n = 0..524

Cf. A004988, A049382, A004994, A216702, A216703, A216704.

seq(product(81-9/k, k=1.. n), n=0..20);
seq((9^n/n!)*product(9*k+8, k=0.. n-1), n=0..20);

Table[Product[81-9/k,{k,n}],{n,0,20}] (* Harvey P. Dale, Jul 20 2021 *)

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 81^n * Gamma(n+8/9) / (Gamma(8/9) * Gamma(n+1)).
a(n) ~ c * 81^n / n^(1/9), where c = 1/Gamma(8/9) = 0.927851... . (End)

A216706 a(n) = Product_{k=1..n} (100 - 10/k).

Original entry on oeis.org

1, 90, 8550, 826500, 80583750, 7897207500, 776558737500, 76546504125000, 7558967282343750, 747497875698437500, 74002289694145312500, 7332954160601671875000, 727184620926332460937500, 72159089307305298046875000, 7164366724082454591796875000

Offset: 0

Michel Lagneau, Sep 16 2012

nonn

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A340725.

seq(product(100-10/k, k=1.. n), n=0..20);
seq((10^n/n!)*product(10*k+9, k=0.. n-1), n=0..20);

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 100^n * Gamma(n+9/10) / (Gamma(9/10) * Gamma(n+1)).
a(n) ~ c * 100^n / n^(1/10), where c = 1/Gamma(9/10) = 1/A340725 = 0.935778... . (End)

A216786 a(n) = Product_{k=1..n} (121 - 11/k).

Original entry on oeis.org

1, 110, 12705, 1490720, 176277640, 20941783632, 2495562549480, 298041470195040, 35653210872081660, 4270462368900447720, 512028438031163681628, 61443412563739641795360, 7378329792029068652259480, 886534702703800402679177520, 106574136046464005550646840440

Offset: 0

Michel Lagneau, Sep 16 2012

nonn

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A216706.

seq(product(121-11/k, k=1.. n), n=0..20);
seq((11^n/n!)*product(11*k+10, k=0.. n-1), n=0..20);
A216786 := proc(n)
    binomial(-10/11,n)*(-121)^n ;
end proc: # R. J. Mathar, Sep 17 2012

Join[{1},FoldList[Times,121-11/Range[20]]] (* Harvey P. Dale, Mar 15 2016 *)

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 121^n * Gamma(n+10/11) / (Gamma(10/11) * Gamma(n+1)).
a(n) ~ c * 121^n / n^(1/11), where c = 1/Gamma(10/11) = 0.942148... . (End)

A283151 Triangle read by rows: Riordan array (1/(1-9x)^(2/3), x/(9x-1)).

Original entry on oeis.org

1, 6, -1, 45, -15, 1, 360, -180, 24, -1, 2970, -1980, 396, -33, 1, 24948, -20790, 5544, -693, 42, -1, 212058, -212058, 70686, -11781, 1071, -51, 1, 1817640, -2120580, 848232, -176715, 21420, -1530, 60, -1, 15677145, -20902860, 9754668, -2438667, 369495, -35190, 2070, -69, 1, 135868590, -203802885

Offset: 0

Tom Richardson, Mar 01 2017

This is an example of a Riordan group involution.
Dual Riordan array of A283150.
With A283150 and A248324, forms doubly infinite Riordan array. For b and c the sequences A283150 and A248324, respectively, and i,j >= 0, the doubly infinite array with d(i,j) = a(i,j), d(-j,-i) = b(i,j), d(i,-j) = c(i,j), and d(-i,j) = 0 (except d(0,0)=1) is a doubly infinite Riordan array.

			Triangle begins
         1;
         6,        -1;
        45,       -15,       1;
       360,      -180,      24,       -1;
      2970,     -1980,     396,      -33,      1;
     24948,    -20790,    5544,     -693,     42,     -1;
    212058,   -212058,   70686,   -11781,   1071,    -51,    1;
   1817640,  -2120580,  848232,  -176715,  21420,  -1530,   60,  -1;
  15677145, -20902860, 9754668, -2438667, 369495, -35190, 2070, -69, 1;

Peter Bala, A 4-parameter family of embedded Riordan arrays
Peter Bala, A note on the diagonals of a proper Riordan Array
H. Prodinger, Some information about the binomial transform, The Fibonacci Quarterly, 32, 1994, 412-415.
Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.

Cf. A004988, A248324, A283150, A025748, A046521.

a(m,n) = binomial(-n-2/3, m-n)*(-1)^m*9^(m-n).
G.f.: (1-9x)^(1/3)/(xy-9x+1).
Recurrence: a(m,n) = a(m,n-1)*(n-1-m)/(9*n-3) for 0 < n <= m; matrix inverse of a(m,n) is a(m,n). - Werner Schulte, Aug 05 2017
From Peter Bala, Mar 05 2018 (Start):
Let P(n,x) = Sum_{k = 0..n} T(n,k)*x^(n-k) denote the n-th row polynomial in descending powers of x. Then (-1)^n*P(n,x) is the n-th degree Taylor polynomial of (1 - 9*x)^(n-1/3) about 0. For example, for n = 4 we have (1 - 9*x)^(11/3) = 2970*x^4 - 1980*x^3 + 396*x^2 - 33*x + 1 + O(x^5).
Let R(n,x) denote the n-th row polynomial of this triangle. The polynomial R(n,9*x) has the e.g.f. Sum_{k = 0..n} T(n,k)*(9*x)^k/k!. The e.g.f. for the n-th diagonal of the triangle (starting at n = 0 for the main diagonal) equals exp(-x) * the e.g.f. for the polynomial R(n,9*x). For example, when n = 3 we have exp(-x)*(360 - 180*(9*x) + 24*(9*x)^2/2! - (9*x)^3/3!) = 360 - 1980*x + 5544*x^2/2! - 11781*x^3/3! + 21420*x^4/4! - ....
Let F(x) = (1 - ( 1 - 9*x)^(1/3))/(3*x). See A025748. The derivatives of F(x) are related to the row polynomials P(n,x) by the identity x^n/n! * (d/dx)^n(F(x)) = 1/(3*x)*( (-1)^n - P(n,x)/(1 - 9*x)^(n-1/3) ), n = 0,1,2,.... Cf. A283151 and A046521. (End)
From Peter Bala, Aug 18 2021: (Start)
T(n,k) = (-1)^k*binomial(n-1/3, n-k)*9^(n-k).
Analogous to the binomial transform we have the following sequence transformation formula: g(n) = Sum_{k = 0..n} T(n,k)*b^(n-k)*f(k) iff f(n) = Sum_{k = 0..n} T(n,k)*b^(n-k)*g(k). See Prodinger, bottom of p. 413, with b replaced with 9*b, c = -1 and d = 2/3.
Equivalently, if F(x) = Sum_{n >= 0} f(n)*x^n and G(x) = Sum_{n >= 0} g(n)*x^n are a pair of formal power series then
G(x) = (1/(1 - 9*b*x)^(2/3)) * F(x/(1 - 9*b*x)) iff F(x) = (1/(1 + 9*b*x)^(2/3)) * G(x/(1 + 9*b*x)).
The infinitesimal generator of the unsigned array has the sequence (9*n+6) n>=0 on the main subdiagonal and zeros elsewhere. The m-th power of the unsigned array has entries m^(n-k)*|T(n,k)|. (End)

Offset corrected by Werner Schulte, Aug 05 2017

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

A097187 Antidiagonal 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, 1, 7, 10, 58, 94, 499, 868, 4360, 7951, 38407, 72508, 339997, 659380, 3019639, 5984968, 26880052, 54249628, 239683171, 491235070, 2139947788, 4444675456, 19125212575, 40190140696, 171064560433, 363227946394, 1531088393647

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.

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

seq(coeff(series(3*x/((1-9*x^2) +(3*x-1)*(1-9*x^2)^(2/3)), x, n+2), x, n), n = 0..30); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[3*x/((1-9*x^2) +(3*x-1)*(1-9*x^2)^(2/3)), {x, 0, 30}], x] (* G. C. Greubel, Sep 17 2019 *)

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

def A097187_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P(3*x/((1-9*x^2) + (3*x-1)*(1-9*x^2)^(2/3))).list()
A097187_list(30) # G. C. Greubel, Sep 17 2019

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

G.f.: A(x) = A004988(x^2)/(1 - x*A097188(x^2)).

A158111 E.g.f.: sm^-1(x) = Sum_{n>=0} a(n)*x^(3n+1)/(3n+1)!; a(n) = coefficient of x^(3n+1)/(3n+1)! in the Maclaurin expansion of the inverse of the Dixon elliptic function sm(x,0).

Original entry on oeis.org

1, 4, 400, 179200, 216832000, 552487936000, 2554704216064000, 19415752042086400000, 225960522265801523200000, 3818732826292045742080000000, 89923520593525093134499840000000, 2854532237720860556461562920960000000, 118891267701073842176624095657984000000000

Offset: 0

Paul D. Hanna, Mar 18 2009

sm(x) = sm(x,0) satisfies: Integral_{y=0..sm(x,0)} dy/(1-y^3)^(2/3) = x.

			E.g.f.: 1/(1-x^3)^(2/3) = 1 + 4*x^3/3! + 400*x^6/6! + 179200*x^9/9! + ...
E.g.f.: sm^-1(x) = x + 4*x^4/4! + 400*x^7/7! + 179200*x^10/10! + ...
sm(x) = x - 4*x^4/4! + 160*x^7/7! - 20800*x^10/10! + 6476800*x^13/13! + ...

Cf. A104133, A004988, A255406.

a(n):= mul(k-0^(mod(k,3)),k=1..3*n):seq(a(n), n = 0 .. 12);
# Peter Bala, Feb 22 2015

Join[{1},Table[Product[(3k-2)(3k-1)^2,{k,n}],{n,14}]] (* Harvey P. Dale, May 19 2012 *)
a[k_] := Pochhammer[2/3, k] (3 k)!/k!; Array[a, 15, 0] (* Jan Mangaldan, Jan 06 2017 *)

PARI
```
a(n)=prod(k=1,n,(3*k-2)*(3*k-1)^2)
```

a(n) = Product_{k=1..n} (3k-2)*(3k-1)^2 for n > 0 with a(0)=1.
E.g.f.: Sum_{n>=0} a(n)*x^(3m)/(3m)! = 1/(1-x^3)^(2/3).
From Peter Bala, Feb 22 2015: (Start)
a(n) = (n - 1/3)! * (3*n)!/( (-1/3)! * n! ).
a(n) = Product {k = 1..3*n} (k - 0^(k mod 3)), where we apply the usual convention that 0^0 = 1. Cf. A255406. (End)
a(n) ~ Gamma(1/3) * 3^(3*n + 1) * n^(3*n + 1/6) / (sqrt(2*Pi) * exp(3*n)). - Vaclav Kotesovec, Apr 10 2018

More terms from Harvey P. Dale, May 19 2012

A216787 a(n) = Product_{k=1..n} (144 - 12/k).

Original entry on oeis.org

1, 132, 18216, 2550240, 359583840, 50917071744, 7230224187648, 1028757612985344, 146597959850411520, 20914642271992043520, 2986610916440463814656, 426813850967673556058112, 61034380688377318516310016, 8732611390798600956948971520, 1250010944797171165551838494720

Offset: 0

Michel Lagneau, Sep 16 2012

nonn

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Cf. A004988, A049382, A004994, A216702, A216703, A216704, A216705, A216706, A216786.

seq(product(144-12/k, k=1.. n), n=0..20);
seq((12^n/n!)*product(12*k+11, k=0.. n-1), n=0..20);

Join[{1},FoldList[Times,144-12/Range[20]]] (* Harvey P. Dale, Dec 22 2015 *)

From Amiram Eldar, Aug 17 2025: (Start)
a(n) = 144^n * Gamma(n+11/12) / (Gamma(11/12) * Gamma(n+1)).
a(n) ~ c * 144^n / n^(1/12), where c = 1/Gamma(11/12) = 0.947376... . (End)

cp's OEIS Frontend

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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A383601 Expansion of 1/( (1-x) * (1-10*x)^2 )^(1/3).

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A216705 a(n) = Product_{k=1..n} (81 - 9/k).

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A216706 a(n) = Product_{k=1..n} (100 - 10/k).

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A216786 a(n) = Product_{k=1..n} (121 - 11/k).

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A283151 Triangle read by rows: Riordan array (1/(1-9x)^(2/3), x/(9x-1)).

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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.

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