A336729 G.f. A(x) satisfies: A(x) = 1 + x * A(x) / (1 + 3 * x * A(x)).

1, 1, -2, 1, 10, -38, 28, 289, -1262, 1054, 11044, -51302, 45604, 482068, -2319176, 2140129, 22753378, -111964106, 105927508, 1130780062, -5652760340, 5444054956, 58291068808, -294808277414, 287740874260, 3088109246572, -15758505143192, 15541351662484, 167103084713608

Offset

0,3

Links

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

Formula

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-3)^(n-k) * binomial(n,k) * binomial(n,k-1) for n > 0.

G.f.: 2/(1 - 4*x + sqrt(1 + 4*x + 16*x^2)).

a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(n,k) * binomial(n+k,n)/(k+1).

(n+1) * a(n) = -2 * (2*n-1) * a(n-1) - 16 * (n-2) * a(n-2) for n>1. - Seiichi Manyama, Aug 08 2020

a(n) ~ 2^(2*n - 1/2) * ((sqrt(3) + 1)*sin(2*Pi*n/3) + (sqrt(3) - 1)*cos(2*Pi*n/3)) / (3^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Dec 04 2020

Mathematica

a[0] = 1; a[n_] := Sum[(-3)^(n - k) * Binomial[n, k] * Binomial[n , k - 1], {k, 1, n}] / n; Array[a, 29, 0] (* Amiram Eldar, Aug 02 2020 *)

Prog

(PARI) {a(n) = local(A=1+x*O(x^n)); for(i=0, n, A=1+x*A/(1+3*x*A)); polcoef(A, n)}
(PARI) {a(n) = if(n==0, 1, sum(k=1, n, (-3)^(n-k)*binomial(n, k)*binomial(n, k-1))/n)}
(PARI) N=40; x='x+O('x^N); Vec(2/(1-4*x+sqrt(1+4*x+16*x^2)))
(PARI) {a(n) = sum(k=0, n, (-3)^k*4^(n-k)*binomial(n, k)*binomial(n+k, n)/(k+1))}

Crossrefs

Column k=3 of A336727.

Cf. A007564, A116091.

Sequence in context: A309234 A071926 A133103 * A304359 A054781 A213421

Adjacent sequences: A336726 A336727 A336728 * A336730 A336731 A336732

Keyword

sign

Author

Seiichi Manyama, Aug 02 2020

Status

approved