A365880 -id:A365880 - cp's OEIS frontend

A365879 Expansion of (1/x) * Series_Reversion( x(1+x)^3(1-x)^5 ).

1, 2, 10, 54, 332, 2162, 14734, 103630, 746857, 5486206, 40926152, 309202686, 2361065920, 18192978966, 141280871840, 1104603758526, 8687878404289, 68692224882620, 545681467048132, 4353096328518810, 34858239962177764, 280095777427596008

Offset: 0

Seiichi Manyama, Sep 21 2023

nonn

Index entries for reversions of series

Cf. A365753, A365856, A365880.
Cf. A365878, A368079.
Cf. A370270.

a(n) = sum(k=0, n, (-1)^k*binomial(3*n+k+2, k)*binomial(6*n-k+4, n-k))/(n+1);

def A365879(n):
    h = binomial(6*n + 4, n) * hypergeometric([-n, 3*(n + 1)], [-6 * n - 4], -1) / (n + 1)
    return simplify(h)
print([A365879(n) for n in range(22)])  # Peter Luschny, Sep 21 2023

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(3*n+k+2,k) * binomial(6*n-k+4,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+k+2,k) * binomial(3*n-2*k+1,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^3 * (1-x)^5 )^(n+1). - Seiichi Manyama, Feb 16 2024

A365854 Expansion of (1/x) * Series_Reversion( x(1+x)^2(1-x)^3 ).

Original entry on oeis.org

1, 1, 4, 13, 55, 232, 1052, 4869, 23206, 112519, 554560, 2767336, 13959941, 71060356, 364569352, 1883143669, 9785481498, 51118097686, 268294595396, 1414106565611, 7481787454031, 39721596068000, 211549545257760, 1129912319370600, 6050931114958080

Offset: 0

Seiichi Manyama, Sep 20 2023

nonn

Index entries for reversions of series

Cf. A365855, A365856.
Cf. A063020, A365878, A365880.
Cf. A365751, A370103.

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

def A365854(n):
    h = binomial(2*(2*n + 1), n) * hypergeometric([-n, 2*(n + 1)], [-2*(2*n + 1)], -1) / (n + 1)
    return simplify(h)
print([A365854(n) for n in range(25)])  # Peter Luschny, Sep 20 2023

a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(2*n+k+1,k) * binomial(4*n-k+2,n-k).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(2*n-2*k,n-2*k). - Seiichi Manyama, Jan 18 2024
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^2 * (1-x)^3 )^(n+1). - Seiichi Manyama, Feb 16 2024

cp's OEIS Frontend

A365879 Expansion of (1/x) * Series_Reversion( x(1+x)^3(1-x)^5 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

SageMath

Formula

A365854 Expansion of (1/x) * Series_Reversion( x(1+x)^2(1-x)^3 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

SageMath

Formula

cp's OEIS Frontend

A365879 Expansion of (1/x) * Series_Reversion( x*(1+x)^3*(1-x)^5 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

SageMath

Formula

A365854 Expansion of (1/x) * Series_Reversion( x*(1+x)^2*(1-x)^3 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

SageMath

Formula

A365879 Expansion of (1/x) * Series_Reversion( x(1+x)^3(1-x)^5 ).

A365854 Expansion of (1/x) * Series_Reversion( x(1+x)^2(1-x)^3 ).