A362210 -id:A362210 - cp's OEIS frontend

A362206 Expansion of 1/(1 - x/(1-9*x)^(1/3)).

1, 1, 4, 25, 181, 1399, 11212, 91936, 765805, 6452449, 54841438, 469306102, 4038193870, 34903997029, 302828905471, 2635745917759, 23003622046900, 201241080558652, 1764149626139119, 15493365042402772, 136288275628625410, 1200600389345625754

Offset: 0

Seiichi Manyama, Apr 11 2023

Cf. A026671, A362157, A362210.

Table[Sum[9^(n - k)*Binomial[n - 2*k/3 - 1, n - k], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Feb 19 2024 *)

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

a(n) ~ (1-9*r)^(1/3) * (6 - 18*r + r^2) / (109 * r^n), where r = 0.1109593191262346... is the root of the equation r*(9 + r^2) = 1. - Vaclav Kotesovec, Feb 19 2024

A373543 Expansion of 1/(1 - x/(1 - 9*x^2)^(2/3)).

Original entry on oeis.org

1, 1, 1, 7, 13, 64, 151, 634, 1693, 6514, 18688, 68239, 204631, 722920, 2230498, 7711216, 24246229, 82612189, 263112874, 887565955, 2852058448, 9553983613, 30892668295, 102975387211, 334454025715, 1110899344549, 3619669508056, 11992016509234, 39164977065622

Offset: 0

Seiichi Manyama, Jun 09 2024

nonn

Cf. A362210, A373544.

Cf. A371456.

a(n) = sum(k=0, n\2, 9^k*binomial(2*n/3-k/3-1, k));

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

a(n) == 1 (mod 3).

A373544 Expansion of 1/(1 - x/(1 - 9*x^3)^(2/3)).

Original entry on oeis.org

1, 1, 1, 1, 7, 13, 19, 70, 157, 280, 799, 1894, 3781, 9646, 23080, 49159, 119203, 283972, 627760, 1487095, 3518617, 7945561, 18620746, 43801447, 100117099, 233475802, 546859390, 1258634440, 2928668632, 6839770279, 15804569341, 36739434904, 85640217781, 198337427839

Offset: 0

Seiichi Manyama, Jun 09 2024

nonn

Cf. A362210, A373543.

Cf. A371458.

a(n) = sum(k=0, n\3, 9^k*binomial(2*n/3-k-1, k));

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

a(n) == 1 (mod 3).

A382516 Expansion of 1/(1 - x/(1 - 9*x)^(4/3)).

Original entry on oeis.org

1, 1, 13, 151, 1693, 18688, 204631, 2230498, 24246229, 263112874, 2852058448, 30892668295, 334454025715, 3619669508056, 39164977065622, 423695451762664, 4583082589819489, 49570596449054509, 536121822834121354, 5798064369702626227, 62702959640721355228

Offset: 0

Seiichi Manyama, Mar 30 2025

Cf. A362206, A362210, A382517.

Cf. A004991.

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

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

D-finite with recurrence (n-1)*(n-2)*a(n) -3*(n-2)*(17*n-35)*a(n-1) +27*(39*n^2-197*n+252)*a(n-2) +2*(-5468*n^2+32199*n-46873)*a(n-3) +6*(9115*n^2-56514*n+77702)*a(n-4) +54*(-1094*n^2-359*n+28901)*a(n-5) +54*(-9846*n^2+134559*n-449254)*a(n-6) +177147*(3*n-19)*(3*n-20)*a(n-7)=0. - R. J. Mathar, Mar 31 2025

A382517 Expansion of 1/(1 - x/(1 - 9*x)^(5/3)).

Original entry on oeis.org

1, 1, 16, 211, 2611, 31426, 373099, 4397527, 51623530, 604629688, 7072089076, 82652922457, 965513250832, 11275328397061, 131649767277064, 1536953772789256, 17941954844917198, 209439428952580837, 2444747948094707815, 28536537876362681194, 333091044353156790346

Offset: 0

Seiichi Manyama, Mar 30 2025

Cf. A362206, A362210, A382516.

Cf. A004992.

a(n) = sum(k=0, n, 9^(n-k)*binomial(n+2*k/3-1, n-k));

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

D-finite with recurrence (n-1)*(n-2)*a(n) -3*(19*n-37)*(n-2)*a(n-1) +27*(49*n^2-233*n+278)*a(n-2) +2*(-7655*n^2+39732*n-47656)*a(n-3) +3*(25519*n^2-98445*n-28306)*a(n-4) +54*(2552*n^2-69623*n+281314)*a(n-5) +27*(-137799*n^2+1870137*n-6193006)*a(n-6) +177147*(99*n^2-1323*n+4418)*a(n-7) -3188646*(3*n-20)*(3*n-22)*a(n-8)=0. - R. J. Mathar, Apr 02 2025

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A362206 Expansion of 1/(1 - x/(1-9*x)^(1/3)).

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A373543 Expansion of 1/(1 - x/(1 - 9*x^2)^(2/3)).

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A373544 Expansion of 1/(1 - x/(1 - 9*x^3)^(2/3)).

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A382516 Expansion of 1/(1 - x/(1 - 9*x)^(4/3)).

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A382517 Expansion of 1/(1 - x/(1 - 9*x)^(5/3)).

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