A049128 -id:A049128 - 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).

A366053 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x+x^3) ).

Original entry on oeis.org

1, 3, 15, 92, 628, 4579, 34917, 275041, 2220472, 18275896, 152780718, 1293657534, 11072033677, 95629771059, 832460471465, 7296161486583, 64331378963164, 570228657335744, 5078345448484216, 45418278349485960, 407749837317844851, 3673300856466182388

Offset: 0

Seiichi Manyama, Sep 27 2023

nonn

Cf. A049128, A114997, A366051, A366052.

a(n) = sum(k=0, n\3, 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)} binomial(n+1,k) * binomial(4*n-2*k+2,n-3*k).

A049133 Revert transform of (x - 1)^2/(1 - x - x^3).

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 15, -31, -308, -1520, -6138, -22260, -74745, -234503, -684931, -1828743, -4249668, -7308296, -592722, 75389838, 487028178, 2286167634, 9278268220, 34247910114, 117081254935, 371845391419, 1086709633580, 2836930639816, 6075557104011

Offset: 1

Olivier Gérard

sign

Index entries for reversions of series

Cf. A049128.

Rest[CoefficientList[InverseSeries[Series[x*(x - 1)^2/(1 - x - x^3), {x, 0, 40}], x], x]] (* Vaclav Kotesovec, Jan 02 2021 *)

Recurrence: 2*n*(2*n - 1)*(26*n^2 - 125*n + 156)*a(n) = 3*(182*n^4 - 1291*n^3 + 3244*n^2 - 3388*n + 1158)*a(n-1) - 3*(78*n^4 - 973*n^3 + 3964*n^2 - 6793*n + 4431)*a(n-2) - (n-3)*(1690*n^3 - 10179*n^2 + 19241*n - 12657)*a(n-3) - 31*(n-4)*(n-3)*(26*n^2 - 73*n + 57)*a(n-4). - Vaclav Kotesovec, Jan 02 2021

a(n+1) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (-1)^k * binomial(n+1,k) * binomial(2*n-2*k,n-3*k). - Seiichi Manyama, Sep 27 2023

A372414 Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x)^2 )^n.

Original entry on oeis.org

1, 1, 3, 13, 55, 231, 981, 4215, 18271, 79735, 349843, 1541783, 6820045, 30263689, 134658681, 600578373, 2684116863, 12017803439, 53895617379, 242054324055, 1088530440315, 4900978877115, 22089865194543, 99662269990363, 450049706481181, 2033999993960581

Offset: 0

Seiichi Manyama, Apr 29 2024

nonn

Cf. A372413, A372415.

Cf. A049128.

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

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x)^2 / (1-x+x^3) ).

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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A366053 Expansion of (1/x) * Series_Reversion( x*(1-x)^4/(1-x+x^3) ).

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A049133 Revert transform of (x - 1)^2/(1 - x - x^3).

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A372414 Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x)^2 )^n.

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