A366053 -id:A366053 - cp's OEIS frontend

A366052 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1-x+x^3) ).

1, 2, 7, 31, 154, 819, 4559, 26226, 154664, 930040, 5680920, 35150493, 219850505, 1387717660, 8828668582, 56553846890, 364449091112, 2361118198094, 15369247139879, 100468188756849, 659271433474584, 4341140182940382, 28675590236716905

Offset: 0

Seiichi Manyama, Sep 27 2023

nonn

Cf. A049128, A114997, A366051, A366053.

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

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

A366051 Expansion of (1/x) * Series_Reversion( x/(1-x+x^3) ).

Original entry on oeis.org

1, -1, 1, 0, -3, 9, -16, 13, 29, -157, 391, -562, -32, 3002, -10373, 20747, -18083, -47941, 271117, -712216, 1066699, 122131, -6464446, 22907125, -46951992, 40883304, 120187926, -679375906, 1809757015, -2731745887, -468147579, 17768126376, -63256877763

Offset: 0

Seiichi Manyama, Sep 27 2023

sign

Cf. A049128, A114997, A366052, A366053.

Cf. A071879.

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

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

A366085 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x-x^3) ).

Original entry on oeis.org

1, 3, 15, 90, 596, 4199, 30869, 234091, 1817720, 14380288, 115492518, 939163680, 7717237661, 63979604459, 534498370665, 4495171005567, 38026764744348, 323358790454352, 2762410748226232, 23697028402783512, 204044822552956179, 1762917281476448944

Offset: 0

Seiichi Manyama, Sep 28 2023

nonn

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

Cf. A049133, A243157, A366084.

Cf. A366053.

a(n) = sum(k=0, n\3, (-1)^k*binomial(n+1, k)*binomial(4*n-2*k+2, n-3*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+1,k) * binomial(4*n-2*k+2,n-3*k).

cp's OEIS Frontend

A366052 Expansion of (1/x) * Series_Reversion( x*(1-x)^3/(1-x+x^3) ).

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A366051 Expansion of (1/x) * Series_Reversion( x/(1-x+x^3) ).

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PARI

Formula

A366085 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x-x^3) ).

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