A389694 Expansion of (1/x) * Series_Reversion( x * (1 - x^2 / (1 - x)^3)^2 ).

1, 0, 2, 6, 23, 98, 425, 1912, 8815, 41400, 197476, 953960, 4657569, 22946580, 113935443, 569564216, 2864247790, 14479892996, 73545875462, 375127824344, 1920653059926, 9867635485890, 50855543471767, 262850098066432, 1362136275216620, 7075962712028884, 36840311644658550

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

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

a(n) = (1/(n+1)) * [x^n] 1/(1 - x^2 / (1 - x)^3)^(2*(n+1)).

Mathematica

Table[(1/(n+1))*Sum[Binomial[2*n+k+1, k]* Binomial[n+k-1, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 31 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x^2/(1-x)^3)^2)/x)
(Magma) [(1/(n+1))*&+[Binomial(2*n+k+1, k) * Binomial(n+k-1, n-2*k): k in [0..Floor(n/2)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 31 2025

Crossrefs

Cf. A368957, A389693.

Cf. A389345.

Sequence in context: A391522 A395133 A280768 * A370183 A278301 A242586

Adjacent sequences: A389691 A389692 A389693 * A389695 A389696 A389697

Keyword

nonn

Author

Seiichi Manyama, Oct 11 2025

Status

approved