A111989 -id:A111989 - cp's OEIS frontend

A111990 Convolution of A111989 with itself.

1, 12, 108, 848, 6192, 43200, 292224, 1933056, 12572928, 80702464, 512532480, 3226742784, 20166803456, 125262102528, 773910872064, 4759428268032, 29151365234688, 177913041518592, 1082361265782784, 6565932190138368

Offset: 0

Wolfdieter Lang, Sep 12 2005

Harvey P. Dale, Table of n, a(n) for n = 0..1000
W. Lang, Riccati meets Fibonacci, The Fibonacci Quarterly, 42 (2004) pp. 231-244, eqs.(58),(59).
Index entries for linear recurrences with constant coefficients, signature (12, -36, -16, 96, 0, -64).

Cf. A111991 (second convolution of A111989).

CoefficientList[Series[1/(1-6x+8x^3)^2,{x,0,30}],x] (* or *) LinearRecurrence[{12,-36,-16,96,0,-64},{1,12,108,848,6192,43200},30] (* Harvey P. Dale, Dec 07 2022 *)

Vec(1/(1-6*x+8*x^3)^2 + O(x^30)) \\ Michel Marcus, Mar 11 2016

G.f.: 1/(1-6*x+8*x^3)^2.

a(n) = (2*(n+1)*b(n+1)-(n+3)*b(n)-4*(n+2)*b(n-1))/9, with b(n):=A111989(n).

A111991 Second convolution of A111989.

Original entry on oeis.org

1, 18, 216, 2136, 18864, 154656, 1203072, 9000576, 65339136, 463112704, 3219204096, 22019254272, 148577427456, 990973034496, 6543595044864, 42832379117568, 278217855664128, 1794871415144448, 11508930723708928

Offset: 0

Wolfdieter Lang, Sep 12 2005

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).

Cf. A111990 (first convolution of A111989), A112003 (third convolution).

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 *)

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

cp's OEIS Frontend

A111990 Convolution of A111989 with itself.

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