A167423 Hankel transform of a simple Catalan convolution.

1, -1, -11, -50, -186, -631, -2029, -6299, -19075, -56704, -166164, -481391, -1381691, -3935125, -11134331, -31328366, -87721614, -244588519, -679429225, -1881102959, -5192705779, -14296088956, -39263958696, -107601905375, -294291714551, -803416991401

Offset: 0

Author

Paul Barry, Nov 03 2009

Keywords

• easy
• sign

Comments

Hankel transform of A167422.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Crossrefs

Cf. A167422, A000045, A000032.

Programs

Magma

[Fibonacci(2*n)*(1-3*n)/2 + Lucas(2*n)*(1-n)/2: n in [0..30]]; // Vincenzo Librandi, Jun 13 2016

Mathematica

Table[((1-3n) Fibonacci[2n] + (1-n) LucasL[2n])/2, {n, 0, 20}] (* Vladimir Reshetnikov, Oct 28 2015 *)
LinearRecurrence[{6, -11, 6, -1}, {1, -1, -11, -50}, 50] (* G. C. Greubel, Jun 12 2016 *)

PARI

Vec((1-7*x+6*x^2-x^3)/(1-6*x+11*x^2-6*x^3+x^4) + O(x^100)) \\ Altug Alkan, Oct 29 2015

Formula

G.f.: ( 1-7*x+6*x^2-x^3 ) / (x^2-3*x+1)^2 .

a(n) = F(2*n)*(1-3*n)/2 + L(2*n)*(1-n)/2. - Paul Barry, Feb 22 2010

a(n) = 3*A001871(n-1) - 2*A001871(n) + F(2*n+4). - Ralf Stephan, May 21 2014

a(n) = 1 - Sum_{k=1..n} k*F(2*k+1), where F(n) = A000045(n). - Vladimir Reshetnikov, Oct 28 2015