A393954 Expansion of ( 1 + Series_Reversion( x*(1-2*x-x^2) ) )^2.

1, 2, 5, 22, 122, 758, 5047, 35208, 253998, 1879438, 14185171, 108782518, 845210808, 6639273704, 52638802244, 420681902896, 3385389615846, 27409394579262, 223110429785275, 1824795182575650, 14988782110022490, 123592674062444070, 1022669887394585295

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

G.f.: 2 - x / Series_Reversion( x*(1-2*x-x^2) ).

G.f.: B(x)^2, where B(x) is the g.f. of A192945.

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

a(n) ~ sqrt(77 - 19*sqrt(7)) * (17 + 7*sqrt(7))^n / (9 * sqrt(21*Pi) * n^(3/2) * 2^(2*n-1)). - Vaclav Kotesovec, Mar 18 2026

Mathematica

Join[{1, 2}, Table[1/(n-1) * Sum[2^(n - 2*k)*Binomial[2*n-2-k, n-2] * Binomial[n-k, k], {k, 0, n/2}], {n, 2, 30}]] (* Vaclav Kotesovec, Mar 18 2026 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec((1+serreverse(x*(1-2*x-x^2)))^2)

Crossrefs

Cf. A120591, A192945, A393955.

Sequence in context: A066305 A020093 A329260 * A381568 A111542 A342432

Adjacent sequences: A393951 A393952 A393953 * A393955 A393956 A393957

Keyword

nonn

Author

Seiichi Manyama, Mar 04 2026

Status

approved