A361840 -id:A361840 - cp's OEIS frontend

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

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

A361845 Expansion of 1/(1 - 9x(1-x)^3)^(1/3).

Original entry on oeis.org

1, 3, 9, 27, 78, 207, 459, 567, -1926, -20763, -120123, -569349, -2410200, -9379449, -33818715, -112292001, -335018295, -837341388, -1317232530, 2358000072, 35974607355, 228270292803, 1148026536963, 5094839173779, 20667058966044, 77501033284779

Offset: 0

Seiichi Manyama, Mar 26 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..1669

Column k=3 of A361840.

Cf. A361816.

a[n_]:=(-9)^n*Binomial[-1/3, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 2/3-n, 2/3-n}, 2^8/3^5]; Array[a,26,0] (* Stefano Spezia, Jul 11 2024 *)

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

n*a(n) = 3 * ( (3*n-2)*a(n-1) - 3*(3*n-4)*a(n-2) + 3*(3*n-6)*a(n-3) - (3*n-8)*a(n-4) ) for n > 3.
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(3*k,n-k).
a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 2/3-n, 2/3-n], 2^8/3^5). - Stefano Spezia, Jul 11 2024

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).

A361847 a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(n*k,n-k).

Original entry on oeis.org

1, 3, 12, 27, -75, -444, 4734, 11532, -466782, 1626750, 50347410, -708889296, -2196754992, 179878246239, -1795732735128, -24691325878980, 953903679982809, -7684914725016600, -226465559200630566, 7742131606464606525, -58889021552013912990

Offset: 0

Seiichi Manyama, Mar 27 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..468

Main diagonal of A361840.

Cf. A361835, A361846.

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

a(n) = [x^n] 1/(1 - 9*x*(1-x)^n)^(1/3).

cp's OEIS Frontend

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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A361845 Expansion of 1/(1 - 9*x*(1-x)^3)^(1/3).

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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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A361847 a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-1/3,k) * binomial(n*k,n-k).

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

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

A361845 Expansion of 1/(1 - 9x(1-x)^3)^(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).