A383601 -id:A383601 - cp's OEIS frontend

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

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

A383605 Expansion of 1/( (1-x) * (1-x-9*x^2)^2 )^(1/3).

Original entry on oeis.org

1, 1, 7, 13, 64, 160, 661, 1927, 7288, 23044, 83413, 275479, 976198, 3301462, 11584861, 39703783, 138747637, 479200129, 1672353256, 5803085008, 20251472416, 70486033288, 246114881956, 858397066324, 2999541427177, 10477699520329, 36642516789607, 128146441442989

Offset: 0

Seiichi Manyama, May 01 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A383601, A383606.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/( (1-x) * (1-x-9*x^2)^2 )^(1/3))); // Vincenzo Librandi, May 06 2025

CoefficientList[Series[1/((1-x)*(1-x-9*x^2)^2)^(1/3),{x,0,27}],x] (* Stefano Spezia, May 02 2025 *)
Table[Sum[(-9)^k*Binomial[-2/3,k]*Binomial[n-k,k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 06 2025 *)

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

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

a(n) ~ Gamma(1/3) * (1 + sqrt(37))^(n + 4/3) / (Pi * 3^(1/6) * 37^(1/3) * n^(1/3) * 2^(n + 7/3)). - Vaclav Kotesovec, May 02 2025

A383606 Expansion of 1/( (1-x) * (1-x-9*x^3)^2 )^(1/3).

Original entry on oeis.org

1, 1, 1, 7, 13, 19, 70, 166, 307, 853, 2164, 4600, 11491, 29137, 66808, 161692, 403843, 961129, 2316238, 5715742, 13831219, 33450073, 82013692, 199820584, 485389276, 1187152906, 2900334583, 7069398325, 17283884710, 42278723290, 103291322056, 252668924536

Offset: 0

Seiichi Manyama, May 01 2025

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A383601, A383605.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/( (1-x) * (1-x-9*x^3)^2 )^(1/3))); // Vincenzo Librandi, May 06 2025

CoefficientList[Series[1/((1-x)*(1-x-9*x^3)^2)^(1/3),{x,0,31}],x] (* Stefano Spezia, May 02 2025 *)
Table[Sum[(-9)^k*Binomial[-2/3,k]*Binomial[n-2*k,k],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 06 2025 *)

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

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

A383610 Expansion of 1/( (1-x^2) * (1-x^2-9*x)^2 )^(1/3).

Original entry on oeis.org

1, 6, 46, 372, 3106, 26406, 227179, 1970952, 17206552, 150940848, 1329193288, 11741662152, 103992267826, 923052335316, 8208568670644, 73116321077784, 652195543067596, 5824848557238228, 52080340709333998, 466116121318516872, 4175438344430632696

Offset: 0

Seiichi Manyama, May 02 2025

nonn

Cf. A383601, A383611.

Cf. A383598.

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

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

a(n) ~ Gamma(1/3) * (9 + sqrt(85))^(n+1) / (Pi * 3^(1/6) * 85^(1/3) * n^(1/3) * 2^(n+2)). - Vaclav Kotesovec, May 03 2025

A383611 Expansion of 1/( (1-x^3) * (1-x^3-9*x)^2 )^(1/3).

Original entry on oeis.org

1, 6, 45, 361, 2982, 25083, 213499, 1832508, 15827103, 137356597, 1196642427, 10457750151, 91630781245, 804632867643, 7078961780064, 62380210284379, 550478616300900, 4863816606663882, 43022548851457447, 380930792260360182, 3375853250109410583

Offset: 0

Seiichi Manyama, May 02 2025

nonn

Cf. A383601, A383610.

Cf. A383599, A383606.

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

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

cp's OEIS Frontend

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

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A383605 Expansion of 1/( (1-x) * (1-x-9*x^2)^2 )^(1/3).

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A383606 Expansion of 1/( (1-x) * (1-x-9*x^3)^2 )^(1/3).

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A383610 Expansion of 1/( (1-x^2) * (1-x^2-9*x)^2 )^(1/3).

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A383611 Expansion of 1/( (1-x^3) * (1-x^3-9*x)^2 )^(1/3).

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