A004987 -id:A004987 - cp's OEIS frontend

A216316 G.f.: 1/( (1-8x)(1+x)^2 )^(1/3).

1, 2, 13, 80, 538, 3740, 26650, 193160, 1417945, 10511450, 78533629, 590485208, 4463274232, 33886781840, 258260802232, 1974759985952, 15143163422794, 116417053435316, 896996316176290, 6925241271855296, 53562550587963052, 414948608904171464, 3219356873886333676

Offset: 0

Paul D. Hanna, Sep 03 2012

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 80*x^3 + 538*x^4 + 3740*x^5 + 26650*x^6 +...
where 1/A(x)^3 = 1 - 6*x - 15*x^2 - 8*x^3.
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 22*x^2/2 + 170*x^3/3 + 1366*x^4/4 + 10922*x^5/5 + 87382*x^6/6 +...+ A007613(n)*x^n/n +...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A004987, A007613, A216317, A216357, A216358.

CoefficientList[Series[1/((1-8*x)*(1+x)^2)^(1/3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

{a(n)=polcoeff(1/( (1-8*x)*(1+x)^2 +x*O(x^n) )^(1/3),n)}

{a(n)=local(A=1+x); A=exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, 3*j))*x^m/m +x*O(x^n))); polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

G.f.: exp( Sum_{n>=1} A007613(n)*x^n/n ), where A007613(n) = Sum_{k=0..n} binomial(3*n,3*k).
Recurrence: n*a(n) = (7*n-5)*a(n-1) + 8*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ Gamma(2/3)*2^(3*n+1)/(3^(5/6)*Pi*n^(2/3)). - Vaclav Kotesovec, Oct 20 2012
Inverse binomial transform of A004987. - Peter Bala, Jul 02 2023

A361840 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9x(1 - x)^k)^(1/3).

Original entry on oeis.org

1, 1, 3, 1, 3, 18, 1, 3, 15, 126, 1, 3, 12, 90, 945, 1, 3, 9, 57, 585, 7371, 1, 3, 6, 27, 297, 3969, 58968, 1, 3, 3, 0, 78, 1629, 27657, 480168, 1, 3, 0, -24, -75, 207, 9216, 196290, 3961386, 1, 3, -3, -45, -165, -438, 459, 53217, 1411965, 33011550

Offset: 0

Seiichi Manyama, Mar 26 2023

			Square array begins:
     1,    1,    1,   1,    1,    1, ...
     3,    3,    3,   3,    3,    3, ...
    18,   15,   12,   9,    6,    3, ...
   126,   90,   57,  27,    0,  -24, ...
   945,  585,  297,  78,  -75, -165, ...
  7371, 3969, 1629, 207, -438, -444, ...

Columns k=0..3 give A004987, A361843, A361844, A361845.
Main diagonal gives A361847.
Cf. A361834, A361839.

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

n*T(n,k) = 3 * Sum_{j=0..k} (-1)^j * binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.

T(n,k) = (-1)^n * Sum_{j=0..n} 9^j * binomial(-1/3,j) * binomial(k*j,n-j).

A361880 Expansion of 1/(1 - 9*x/(1 - x)^2)^(1/3).

Original entry on oeis.org

1, 3, 24, 207, 1893, 17952, 174402, 1723494, 17250000, 174354822, 1776119970, 18208500000, 187659221409, 1942674634371, 20187543581880, 210472842939975, 2200677521078253, 23068297001178240, 242353695578011416, 2551260130246575048, 26905595698893121728

Offset: 0

Seiichi Manyama, Mar 28 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..961

Cf. A004987, A361375, A361843, A361844, A361845, A361895, A361896.

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

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n+k-1,n-k).
a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (n+2*k) * (n-k) * a(k).
(n-1)*n*a(n) = (11*n-6)*(n-1)*a(n-1) - 18*(n-2)*a(n-2) - (11*n-38)*(n-3)*a(n-3) + (n-3)*(n-4)*a(n-4) for n > 3.
a(n) ~ 3^(1/3) * ((11 + 3*sqrt(13))/2)^n / (Gamma(1/3) * 13^(1/6) * n^(2/3)). - Vaclav Kotesovec, Mar 28 2023

A361895 Expansion of 1/(1 - 9*x/(1 - x)^3)^(1/3).

Original entry on oeis.org

1, 3, 27, 252, 2487, 25434, 266364, 2837082, 30601233, 333302931, 3658565127, 40413860334, 448778693844, 5005642415907, 56044616215041, 629552293867800, 7092072533703567, 80095810435943526, 906605837653876254, 10282430320166723448, 116829834042508121682

Offset: 0

Seiichi Manyama, Mar 28 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..932

Cf. A004987, A361375, A361843, A361844, A361845, A361880, A361896.

a[0]=1; a[n_]:=3*n*(1 + n)*HypergeometricPFQ[{1-n, 1+n/2, (3+n)/2}, {5/3, 2}, -4/3]/2; Array[a,21,0] (* Stefano Spezia, May 02 2024 *)

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

a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(n+2*k-1,n-k).
a(0) = 1; a(n) = (3/n) * Sum_{k=0..n-1} (n+2*k) * binomial(n+1-k,2) * a(k).
a(n) = 3*n*(1 + n)*hypergeom([1-n, 1+n/2, (3+n)/2], [5/3, 2], -4/3)/2 for n > 0. - Stefano Spezia, May 02 2024
a(n) ~ ((7 - sqrt(21))^(1/3) + (7 + sqrt(21))^(1/3))^(1/3) * (4 + (3*((39 - sqrt(21))/2))^(1/3) + (3*((39 + sqrt(21))/2))^(1/3))^n / (Gamma(1/3) * 2^(1/9) * 7^(2/9) * n^(2/3)). - Vaclav Kotesovec, Jul 11 2025

A224881 Expansion of 1/(1 - 16*x)^(1/8).

Original entry on oeis.org

1, 2, 18, 204, 2550, 33660, 460020, 6440280, 91773990, 1325624300, 19354114780, 285033326760, 4227994346940, 63094684869720, 946420273045800, 14259398780556720, 215673406555920390, 3273161111260438860, 49824785804742235980, 760483572809223601800

Offset: 0

Paul D. Hanna, Jul 23 2013

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 204*x^3 + 2550*x^4 + 33660*x^5 + ...
where
A(x)^8 = 1 + 16*x + 256*x^2 + 4096*x^3 + 65536*x^4 + ... + 16^n*x^n + ...
Also,
A(x)^4 = 1 + 8*x + 96*x^2 + 1280*x^3 + 17920*x^4 + 258048*x^5 + ... + 4^n*A000984(n)*x^n + ...
A(x)^2 = 1 + 4*x + 40*x^2 + 480*x^3 + 6240*x^4 + 84864*x^5 + ... + 2^n*A004981(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), this sequence (b=16), A034688 (b=25), A298799 (b=27), A004993 (b=36), A034835 (b=49).

Cf. A301271.

List([0..20],n->(2^n/Factorial(n))*Product([0..n-1],k->8*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series(1/(1-16*x)^(1/8), x,50),x,n+1),n=0..20); # Muniru A Asiru, Jun 23 2018

CoefficientList[Series[1/(1-16*x)^(1/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)

{a(n)=polcoeff(1/(1-16*x +x*O(x^n))^(1/8),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=(2^n/n!)*prod(k=0,n-1,8*k + 1)}
for(n=0,30,print1(a(n),", "))

a(n) = (2^n/n!) * Product_{k=0..n-1} (8*k + 1).

a(n) ~ 16^n/(GAMMA(1/8)*n^(7/8)). - Vaclav Kotesovec, Jul 24 2013

A298799 Expansion of (1-27*x)^(-1/9).

Original entry on oeis.org

1, 3, 45, 855, 17955, 398601, 9167823, 216098685, 5186368440, 126201632040, 3104560148184, 77049538223112, 1926238455577800, 48452305767226200, 1225151160114148200, 31118839466899364280, 793530406405933789140, 20305042752151835192700

Offset: 0

Seiichi Manyama, Jun 22 2018

nonn

Conjecture: a(p*n) == a(n) (mod p^2) for prime p == 1 (mod 9) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/9 and 1 <= k <= (p-1)/9. Cf. A034171, A004981 and A004982. - Peter Bala, Dec 23 2019

Seiichi Manyama, Table of n, a(n) for n = 0..700

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), A224881 (b=16), A034688 (b=25), this sequence (b=27), A004993 (b=36), A034835 (b=49).

Cf. A305991, A034171, A004981, A004982.

List([0..20],n->(3^n/Factorial(n))*Product([0..n-1],k->9*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series((1-27*x)^(-1/9), x, n+1), x, n), n=0..20); # Muniru A Asiru, Jun 23 2018
# Alternative:
A298799 := n -> (-27)^n*binomial(-1/9, n):
seq(A298799(n), n=0..17); # Peter Luschny, Dec 26 2019

N=20; x='x+O('x^N); Vec((1-27*x)^(-1/9))

a(n) = 3^n/n! * Product_{k=0..n-1} (9*k + 1) for n > 0.
a(n) ~ 3^(3*n) / (Gamma(1/9) * n^(8/9)). - Vaclav Kotesovec, Jun 23 2018
From Peter Luschny, Dec 26 2019: (Start)
a(n) = (-27)^n*binomial(-1/9, n).
a(n) = n! * [x^n] hypergeom([1/9], [1], 27*x). (End)
D-finite with recurrence: n*a(n) +3*(-9*n+8)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9x(1 + x)^k)^(1/3).

Original entry on oeis.org

1, 1, 3, 1, 3, 18, 1, 3, 21, 126, 1, 3, 24, 162, 945, 1, 3, 27, 201, 1341, 7371, 1, 3, 30, 243, 1809, 11529, 58968, 1, 3, 33, 288, 2352, 16893, 101619, 480168, 1, 3, 36, 336, 2973, 23607, 161676, 911466, 3961386, 1, 3, 39, 387, 3675, 31818, 242757, 1574289, 8281737, 33011550

Offset: 0

Seiichi Manyama, Mar 26 2023

			Square array begins:
     1,     1,     1,     1,     1,     1, ...
     3,     3,     3,     3,     3,     3, ...
    18,    21,    24,    27,    30,    33, ...
   126,   162,   201,   243,   288,   336, ...
   945,  1341,  1809,  2352,  2973,  3675, ...
  7371, 11529, 16893, 23607, 31818, 41676, ...

Columns k=0..3 give A004987, A180400, A361841, A361842.
Main diagonal gives A361846.
Cf. A361830, A361840.

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

n*T(n,k) = 3 * Sum_{j=0..k} binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.

T(n,k) = Sum_{j=0..n} (-9)^j * binomial(-1/3,j) * binomial(k*j,n-j).

A361896 Expansion of 1/(1 - 9*x/(1 - x)^4)^(1/3).

Original entry on oeis.org

1, 3, 30, 300, 3165, 34584, 386880, 4400928, 50692266, 589584042, 6910397886, 81507086634, 966408021984, 11509174498254, 137584249375308, 1650109151463594, 19847075122106145, 239316542492974317, 2892135259684291248, 35021199836282568456, 424837125616822551264

Offset: 0

Seiichi Manyama, Mar 28 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..907

Cf. A004987, A361375, A361843, A361844, A361845, A361880, A361895.

my(N=30, x='x+O('x^N)); Vec(1/(1-9*x/(1-x)^4)^(1/3))

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

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

A362157 Expansion of -1/(1 - x * (1-9*x)^(1/3)).

Original entry on oeis.org

-1, -1, 2, 14, 62, 317, 1967, 13544, 98012, 731273, 5581682, 43377617, 342035204, 2729286017, 21996002729, 178769985791, 1463465010995, 12055667829734, 99858109234379, 831150925169516, 6947819513136614, 58303331305857293, 490961900646977570

Offset: 0

Seiichi Manyama, Apr 09 2023

Cf. A004987, A362156.

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

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

A376806 Expansion of 1/((1 - x^3)^3 - 9*x)^(1/3).

Original entry on oeis.org

1, 3, 18, 127, 957, 7497, 60229, 492483, 4079826, 34138873, 287946771, 2444458878, 20863127251, 178868929074, 1539439262406, 13293346718161, 115123998810525, 999553370761017, 8698196146639573, 75845560146124527, 662551474429229571, 5797239969198654748

Offset: 0

Seiichi Manyama, Oct 04 2024

nonn

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

Cf. A376802, A376805.

Cf. A004987.

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

a(n) = Sum_{k=0..floor(n/3)} (-9)^(n-3*k) * binomial(-1/3,n-3*k) * binomial(3*n-8*k,k).

cp's OEIS Frontend

A216316 G.f.: 1/( (1-8*x)*(1+x)^2 )^(1/3).

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A361840 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 - x)^k)^(1/3).

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A361880 Expansion of 1/(1 - 9*x/(1 - x)^2)^(1/3).

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A361895 Expansion of 1/(1 - 9*x/(1 - x)^3)^(1/3).

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A224881 Expansion of 1/(1 - 16*x)^(1/8).

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A298799 Expansion of (1-27*x)^(-1/9).

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A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 + x)^k)^(1/3).

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A361896 Expansion of 1/(1 - 9*x/(1 - x)^4)^(1/3).

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A216316 G.f.: 1/( (1-8x)(1+x)^2 )^(1/3).

A361840 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9x(1 - x)^k)^(1/3).

A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9x(1 + x)^k)^(1/3).