A168491 -id:A168491 - cp's OEIS frontend

A369076 Expansion of (1/x) * Series_Reversion( x * (1+x^2/(1-x))^2 ).

1, 0, -2, -2, 9, 24, -37, -240, -2, 2126, 2919, -16052, -50663, 86940, 631995, 19094, -6491463, -9595434, 54443985, 181532910, -317331187, -2426618056, -133151895, 26332109928, 40544827703, -230619508548, -793966990358, 1384746844832, 10960715925621, 881359815524

Offset: 0

Seiichi Manyama, Jan 12 2024

sign

Index entries for reversions of series

Cf. A168491, A369077.
Cf. A368969, A368973, A368975.
Cf. A368957.

my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1+x^2/(1-x))^2)/x)

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

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

A369077 Expansion of (1/x) * Series_Reversion( x * (1+x^3/(1-x))^2 ).

Original entry on oeis.org

1, 0, 0, -2, -2, -2, 13, 32, 55, -72, -439, -1152, -506, 4870, 20613, 31744, -26392, -313096, -826529, -654362, 3635175, 16431826, 30100349, -15474300, -262654439, -780688624, -756130333, 3013376172, 15711713509, 31584466782, -6090973971, -250819494954

Offset: 0

Seiichi Manyama, Jan 12 2024

sign

Index entries for reversions of series

Cf. A168491, A369076.
Cf. A368970, A368974, A368976.
Cf. A369011.

my(N=40, x='x+O('x^N)); Vec(serreverse(x*(1+x^3/(1-x))^2)/x)

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

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

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

Original entry on oeis.org

1, 3, 1, 10, 6, 1, 35, 30, 9, 1, 126, 140, 60, 12, 1, 462, 630, 350, 100, 15, 1, 1716, 2772, 1890, 700, 150, 18, 1, 6435, 12012, 9702, 4410, 1225, 210, 21, 1, 24310, 51480, 48048, 25872, 8820, 1960, 280, 24, 1

Offset: 0

Philippe Deléham, Dec 19 2009

			Triangle begins:
     1;
     3,    1;
    10,    6,    1;
    35,   30,    9,   1;
   126,  140,   60,  12,   1;
   462,  630,  350, 100,  15,  1;
  1716, 2772, 1890, 700, 150, 18, 1;
  ...

Cf. A107230, A171651

Cf. A001405, A001700, A005573, A005773, A026378, A099323, A122898, A168491.

T[n_,k_]:=n!SeriesCoefficient[Exp[2*x]*(BesselI[0,2*x]+BesselI[1,2*x])*x^k / k!,{x,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Dec 23 2023 *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n+1), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -4, -3, -2, -1, 0, 1, 2, 3 respectively.
Conjectural g.f.: 1/(2*t)*( sqrt( (1 - x*t)/(1 - (4 + x)*t) ) - 1 ) = 1 + (3 + x)*t + (10 + 6*x + x^2)*t^2 + .... - Peter Bala, Nov 10 2013
E.g.f. of column k: exp(2*x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 23 2023

cp's OEIS Frontend

A369076 Expansion of (1/x) * Series_Reversion( x * (1+x^2/(1-x))^2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A369077 Expansion of (1/x) * Series_Reversion( x * (1+x^3/(1-x))^2 ).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A171814 Triangle T : T(n,k)= A007318(n,k)*A001700(n-k).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula