A392640 Expansion of 1 / sqrt((1-x)^4 - 4*x^3).

1, 2, 3, 6, 17, 48, 125, 320, 843, 2274, 6165, 16698, 45317, 123520, 337941, 926804, 2546315, 7007970, 19320269, 53345662, 147489843, 408261820, 1131321795, 3138104148, 8712572219, 24209742230, 67324021713, 187353283062, 521725289085, 1453757004756, 4053153406353

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

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

D-finite with recurrence (n + 2)*a(n) + (-20 - 8*n)*a(n + 1) + (6*n + 18)*a(n + 2) + (-14 - 4*n)*a(n + 3) + (n + 4)*a(n + 4) = 0. - Robert Israel, Jan 18 2026

Maple

f:= gfun:-rectoproc({(n + 2)*a(n) + (-20 - 8*n)*a(n + 1) + (6*n + 18)*a(n + 2) + (-14 - 4*n)*a(n + 3) + (n + 4)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 3, a(3) = 6}, a(n), remember):
map(f, [$0..40]); # Robert Israel, Jan 18 2026

Mathematica

Table[Sum[Binomial[2*k, k]*Binomial[n+k+1, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 36}] (* Vincenzo Librandi, Jan 19 2026 *)

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(1/sqrt((1-x)^4-4*x^3))
(Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R! 1 / Sqrt((1-x)^4 - 4*x^3)); // Vincenzo Librandi, Jan 19 2026

Crossrefs

Cf. A101500, A162480.

Cf. A098479, A392643.

Sequence in context: A338594 A375565 A078344 * A024498 A319283 A325298

Adjacent sequences: A392637 A392638 A392639 * A392641 A392642 A392643

Keyword

nonn

Author

Seiichi Manyama, Jan 18 2026

Status

approved