A389628 Expansion of (1/x) * Series_Reversion( x * (1 - x)^2 * (1 - x / (1 - x)^3) ).

1, 3, 18, 136, 1153, 10480, 99821, 983350, 9936412, 102418152, 1072639570, 11382045819, 122107547247, 1322206171494, 14431844044967, 158618667690449, 1753981473125002, 19499667701984703, 217823525073795357, 2443664670465204376, 27520333306494932680, 311016232016769939900

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Formula

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

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

Mathematica

Table[(1/(n+1))*Sum[Binomial[n+k, k]* Binomial[3*n+2*k+1, n-k], {k, 0, n}], {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/(1-x)^3))/x)
(Magma) [(1/(n+1))*&+[Binomial(n+k, k) * Binomial(3*n+2*k+1, n-k): k in [0..n]] : n in [0..30] ]; // Vincenzo Librandi, Oct 31 2025

Crossrefs

Cf. A006632, A085614, A249924.

Cf. A365150, A378466.

Sequence in context: A247452 A371416 A376224 * A118970 A003122 A275549

Adjacent sequences: A389625 A389626 A389627 * A389629 A389630 A389631

Keyword

nonn

Author

Seiichi Manyama, Oct 09 2025

Status

approved