A004987 -id:A004987 - cp's OEIS frontend

A376282 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^7)^(1/3).

1, 3, 54, 1368, 40365, 1299078, 44223732, 1565864784, 57079952046, 2127818007315, 80742077597610, 3108398557803480, 121107814518484872, 4766365291226837508, 189209375036491438800, 7567095678024459993120, 304603864960375133224533, 12331716699093681951702810

Offset: 0

Seiichi Manyama, Oct 23 2024

nonn

Paolo Xausa, Table of n, a(n) for n = 0..600

Cf. A004987, A008931, A078532, A245114, A376636.

A376282[n_] := 9^n*Binomial[(7*n + 1)/3, n]/(7*n + 1);
Array[A376282, 20, 0] (* Paolo Xausa, Aug 04 2025 *)

a(n) = 9^n*binomial(7*n/3+1/3, n)/(7*n+1);

a(n) = 9^n * binomial(7*n/3 + 1/3,n)/(7*n+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^11). - Seiichi Manyama, Jun 20 2025
D-finite with recurrence 8*n*(n-1)*(n-2)*(4*n-5)*(2*n-1)*(4*n+1)*a(n) -189*(7*n-11)*(7*n-17)*(7*n-2)*(7*n-20)*(7*n-5)*(7*n-8)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

Original entry on oeis.org

1, 4, 40, 480, 6240, 84864, 1188096, 16972800, 246105600, 3609548800, 53421322240, 796463349760, 11946950246400, 180123249868800, 2727580640870400, 41459225741230080, 632253192553758720

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Straub, V. H. Moll, and T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10).

Cf. A007696.

Expansion of (1-b^2*x)^(-1/b): A000984 (b=2), A004987 (b=3), this sequence (b=4), A034688 (b=5), A004993 (b=6), A034835 (b=7), A034977 (b=8), A035024 (b=9), A035308 (b=10).

CoefficientList[Series[1/Surd[1-16x,4],{x,0,20}],x] (* Harvey P. Dale, Aug 06 2018 *)

a(n) = (4^n/n!)*A007696(n), n >= 1, a(0) := 1, A007696(n) = (4*n-3)!^4 := Product_{j = 1..n} 4*j - 3.
G.f.: (1 - 16*x)^(-1/4).
D-finite with recurrence: n*a(n) + 4*(-4*n + 3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-16)^n*binomial(-1/4, n).
a(n) ~ Gamma(3/4)/(sqrt(2)*Pi) * 16^n/n^(3/4).
E.g.f.: hypergeom([1/4], [1], 16*x).
a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/4, k)* binomial(-1/4, 2*n - k).
(16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = (4^n)*binomial(2*n, n) = A098430.
Sum_{k = 0..2*n} a(k)*a(2*n-k) = (16^n)*binomial(4*n, 2*n). (End)

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).

A361843 Expansion of 1/(1 - 9x(1-x))^(1/3).

Original entry on oeis.org

1, 3, 15, 90, 585, 3969, 27657, 196290, 1411965, 10261485, 75183147, 554480316, 4111617510, 30628393110, 229048769790, 1718666596692, 12933847045701, 97584913269675, 737953856289675, 5591915004100950, 42450848142844995, 322796964495941235

Offset: 0

Seiichi Manyama, Mar 26 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..1105

Column k=1 of A361840.

Cf. A004987.

A361843 := n -> (-9)^n*binomial(-1/3, n)*hypergeom([1/2 - n/2, -n/2], [2/3 - n], 4/9): seq(simplify(A361843(n)), n = 0..21); # Peter Luschny, Mar 27 2023

my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1-x))^(1/3))

n*a(n) = 3 * ( (3*n-2)*a(n-1) - (3*n-4)*a(n-2) ) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(k,n-k).
a(n) = A004987(n)*hypergeom([1/2 - n/2, -n/2], [2/3 - n], 4/9). - Peter Luschny, Mar 27 2023
a(n) ~ 3^n * phi^(2*n + 2/3) / (Gamma(1/3) * 5^(1/6) * n^(2/3)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 29 2023

A361844 Expansion of 1/(1 - 9x(1-x)^2)^(1/3).

Original entry on oeis.org

1, 3, 12, 57, 297, 1629, 9216, 53217, 311796, 1846818, 11032416, 66356712, 401364531, 2439135585, 14882263002, 91116281565, 559528781697, 3445002647847, 21260140172244, 131474746842345, 814564464082263, 5055177167348463, 31420067723814780

Offset: 0

Seiichi Manyama, Mar 26 2023

nonn

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

Column k=2 of A361840.

Cf. A361815, A004987.

A361844 := n -> (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3,
-n*2/3], [1/2 - n, 2/3 - n], 3/4):
seq(simplify(A361844(n)), n = 0..22); # Peter Luschny, Mar 27 2023

my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1-x)^2)^(1/3))

n*a(n) = 3 * ( (3*n-2)*a(n-1) - 2*(3*n-4)*a(n-2) + (3*n-6)*a(n-3) ) for n > 2.
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(2*k,n-k).
a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([1/3 - n*2/3, 2/3 - n*2/3, -n*2/3], [1/2 - n, 2/3 - n], 3/4). - Peter Luschny, Mar 27 2023

A124214 E.g.f.: exp(x) / (2 - exp(3*x))^(1/3).

Original entry on oeis.org

1, 2, 10, 98, 1402, 26162, 601930, 16462658, 521659162, 18791451602, 758345497450, 33889063202018, 1661229537252922, 88627461127536242, 5112116659677605770, 317007674364657538178, 21030558126242472270682

Offset: 0

Karol A. Penson, Oct 19 2006

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A229558, A004987, A124212.

A124214 := proc(n)
    exp(x)/root[3](2-exp(3*x)) ;
    coeftayl(%,x=0,n)*n! ;
end proc: # R. J. Mathar, Dec 19 2013

CoefficientList[Series[Exp[x]/(2-Exp[3*x])^(1/3), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)

a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^4+x*O(x^n))); n!*polcoeff(A, n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 18 2013

a(n) ~ Gamma(2/3)*3^(n+1/2)*n^(n-1/6)/(sqrt(2*Pi)*exp(n)*(log(2))^(n+1/3)). - Vaclav Kotesovec, Jun 26 2013
E.g.f. A(x) satisfies: A'(x) = A(x) + A(x)^4. - Paul D. Hanna, Dec 18 2013
E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^3 dx) with A(0)=1. - Paul D. Hanna, Dec 18 2013
a(n) = 2^(-1/3) * Sum_{k >= 0} (1/18)^k*A004987(k)*(3*k + 1)^n = 2^(-1/3) * Sum_{k >= 0} (-1/2)^k*binomial(-1/3, k)*(3*k + 1)^n. Cf. A124212 and A229558. - Peter Bala, Aug 30 2016

A247029 G.f. A(x) satisfies A(x) = A(x)^4 - 9*x.

Original entry on oeis.org

1, 3, -18, 180, -2187, 29484, -424116, 6377292, -99034650, 1576075644, -25569752274, 421325812440, -7031733125508, 118620405322020, -2019349799669160, 34647126360607440, -598525520999144643, 10401492640172342940, -181721630178565389900, 3189811189331825319492

Offset: 0

Paul D. Hanna, Sep 09 2014

sign

			G.f.: A(x) = 1 + 3*x - 18*x^2 + 180*x^3 - 2187*x^4 + 29484*x^5 - 424116*x^6 +...
where
A(x)^4 = 1 + 12*x - 18*x^2 + 180*x^3 - 2187*x^4 + 29484*x^5 - 424116*x^6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A004987, A008931, A078532, A245114, A376282, A376636.

Cf. A224884.

FullSimplify[Table[-(-1)^n * 3^(2*n-1) * 4^(n-1) * Gamma[n/3 + 1/6] * Gamma[2*n/3 - 1/6] / (Pi * Gamma[n + 1]), {n, 0, 20}]] (* Vaclav Kotesovec, Nov 18 2017 *)

{a(n)=polcoeff(x/serreverse(x*(1+9*x +x^2*O(x^n))^(1/3)), n)}
for(n=0, 25, print1(a(n), ", "))

G.f.: x / Series_Reversion( x*(1 + 9*x)^(1/3) ).
Recurrence: (n-2)*(n-1)*n*a(n) = -216*(2*n - 5)*(4*n - 13)*(4*n - 7)*a(n-3). - Vaclav Kotesovec, Nov 18 2017
a(n) ~ -(-1)^n * 2^(8*n/3 - 13/6) * 3^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 18 2017
G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^5). - Seiichi Manyama, Jun 20 2025

A268549 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - z) * (1 - v - w)).

Original entry on oeis.org

1, 12, 648, 50400, 4630500, 468087984, 50345463168, 5655718328832, 656151696743400, 78036148295820000, 9465472643689782720, 1166663950520357802240, 145719568153188579382560, 18405635030728188793200000

Offset: 0

N. J. A. Sloane, Feb 29 2016

nonn

"The corresponding (order-three) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension." (see A. Bostan link) - Gheorghe Coserea, Aug 15 2016

			1 + 12*x + 648*x^2 + 50400*x^3 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..100
A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (30).
Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Cf. A004987, A268545-A268555.

A268549 := proc(n)
    (1-9*x*y)/(1-3*y-2*x+3*y^2+9*x^2*y)/(1-u-z)/(1-v-w) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,n) ;
    coeftayl(%,z=0,n) ;
    coeftayl(%,u=0,n) ;
    coeftayl(%,v=0,n) ;
    coeftayl(%,w=0,n) ;
end proc:
seq(A268549(n),n=0..40) ; # R. J. Mathar, Mar 11 2016
series(hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x), x=0, 14); # Gheorghe Coserea, Aug 15 2016

FullSimplify[Table[3^(2*n)*(2*n)!^2*Gamma[n + 1/3]/(Gamma[1/3]*(n!)^5), {n, 0, 15}]] (* Vaclav Kotesovec, Jul 01 2016 *)

a(n) = [(xyzuvw)^n] (1 - 9*x*y)/((1 - 3*y - 2*x + 3*y^2 + 9*x^2*y) * (1 - u - z) * (1 - v - w)).
D-finite with recurrence: n^3*a(n) -12*(3*n-2)*(-1+2*n)^2*a(n-1)=0. - R. J. Mathar, Mar 11 2016 [follows from the hypergeometric g.f. below - Georg Fischer, Jul 30 2022]
From Vaclav Kotesovec, Jul 01 2016: (Start)
a(n) = 3^(2*n) * (2*n)!^2 * Gamma(n + 1/3) / (Gamma(1/3) * (n!)^5).
a(n) ~ 12^(2*n)/(Gamma(1/3)*Pi*n^(5/3)).
(End)
From Gheorghe Coserea, Aug 16 2016: (Start)
a(n) = [(xyzuv)^n] 1/((1 - x + 3*y - 27*x*y^3 - 27*x*y^2 - 9*x*y + 3*y^2) * (1 - u - v - u*z - v*z)).
G.f.: hypergeom([1/3, 1/2, 1/2], [1, 1], 144*x).
(End)

A376802 Expansion of 1/((1 - x)^3 - 9*x)^(1/3).

Original entry on oeis.org

1, 4, 31, 283, 2770, 28204, 294568, 3131650, 33732883, 367035814, 4025600941, 44439461275, 493218155119, 5498860571026, 61543476786067, 691095770653867, 7783168304357434, 87878978740300960, 994484816394177214, 11276915136560900662, 128106749179069022344

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..936

Partial sums of A361895.
Cf. A098536, A376803, A376804.
Cf. A376805, A376806.
Cf. A004987.

CoefficientList[Series[1/Surd[((1-x)^3-9x),3],{x,0,30}],x] (* Harvey P. Dale, Dec 11 2024 *)

my(N=30, x='x+O('x^N)); Vec(1/((1-x)^3-9*x)^(1/3))

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n+2*k,n-k).

a(n) = hypergeom([(1+n)/2, 1+n/2, -n], [2/3, 1], -4/3). - Stefano Spezia, May 04 2025

A383597 Expansion of 1/( (1-x)^2 * (1-10*x) )^(1/3).

Original entry on oeis.org

1, 4, 25, 190, 1570, 13552, 120178, 1085620, 9940345, 91962460, 857750233, 8053389142, 76026759760, 721017894640, 6864725124520, 65578937628304, 628320730656586, 6035594205744520, 58110220504754650, 560624083417180300, 5418599393597801020, 52459116546784350880

Offset: 0

Seiichi Manyama, May 01 2025

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

Cf. A383598, A383599.

Cf. A004987, A361375, A376802, A383601.

I:=[4,25]; [1] cat [n le 2 select I[n] else ((11*n-7)*Self(n-1) - 10*(n-1) *Self(n-2))/n : n in [1..30]]; // Vincenzo Librandi, May 04 2025

Table[Sum[(-9)^k *Binomial[-1/3,k]* Binomial[n, k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, May 04 2025 *)

a(n) = sum(k=0, n, (-9)^k*binomial(-1/3, k)*binomial(n, k));

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n,k).
n*a(n) = (11*n-7)*a(n-1) - 10*(n-1)*a(n-2) for n > 1.
a(n) ~ 10^(n + 2/3) / (Gamma(1/3) * 3^(4/3) * n^(2/3)). - Vaclav Kotesovec, May 02 2025
a(n) = hypergeom([1/3, -n], [1], -9). - Stefano Spezia, May 04 2025

cp's OEIS Frontend

A376282 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^7)^(1/3).

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A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

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

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A361843 Expansion of 1/(1 - 9*x*(1-x))^(1/3).

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A361844 Expansion of 1/(1 - 9*x*(1-x)^2)^(1/3).

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A124214 E.g.f.: exp(x) / (2 - exp(3*x))^(1/3).

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A247029 G.f. A(x) satisfies A(x) = A(x)^4 - 9*x.

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A268549 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - z) * (1 - v - w)).

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A376282 G.f. A(x) satisfies A(x) = (1 + 9xA(x)^7)^(1/3).

A361843 Expansion of 1/(1 - 9x(1-x))^(1/3).

A361844 Expansion of 1/(1 - 9x(1-x)^2)^(1/3).