A111991 Second convolution of A111989.
1, 18, 216, 2136, 18864, 154656, 1203072, 9000576, 65339136, 463112704, 3219204096, 22019254272, 148577427456, 990973034496, 6543595044864, 42832379117568, 278217855664128, 1794871415144448, 11508930723708928
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (18, -108, 192, 288, -864, -192, 1152, 0, -512).
Programs
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Mathematica
CoefficientList[Series[1/(1-6x+8x^3)^3,{x,0,30}],x] (* or *) LinearRecurrence[{18,-108,192,288,-864,-192,1152,0,-512},{1,18,216,2136,18864,154656,1203072,9000576,65339136},30] (* Harvey P. Dale, Oct 10 2013 *)
Formula
G.f.: 1/(1-6*x+8*x^3)^3.
a(n) = (2*(n+1)*b(n+1)-(n+6)*b(n)-4*(n+5)*b(n-1))/(2*9), with b(n):=A111990(n).
a(n) = ((35*n^2+75*n+54)*c(n)-4*(n+2)*(2*n-1)*c(n-1)-48*(n+1)^2*c(n-2))/54, with c(n):= A111989(n), especially, c(-2) = 0 = c(-1).
a(0)=1, a(1)=18, a(2)=216, a(3)=2136, a(4)=18864, a(5)=154656, a(6)=1203072, a(7)=9000576, a(8)=65339136, a(n)=18*a(n-1)- 108*a(n-2)+ 192*a(n-3)+ 288*a(n-4) -864*a(n-5)-192*a(n-6)+1152*a(n-7)-512*a(n-9). - Harvey P. Dale, Oct 10 2013