A083085 -id:A083085 - cp's OEIS frontend

A083086 a(n) = (2^(n+1) + (-4)^n)/3.

1, 0, 8, -16, 96, -320, 1408, -5376, 22016, -87040, 350208, -1396736, 5595136, -22364160, 89489408, -357892096, 1431699456, -5726535680, 22906667008, -91625619456, 366504574976, -1466014105600, 5864064811008, -23456242466816, 93825003421696, -375299946577920

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A083085.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,8).

Cf. A000079, A078008, A083085, A083216.

[(2*2^n+(-4)^n)/3: n in [0..30]]; // Vincenzo Librandi, Nov 12 2011

LinearRecurrence[{-2,8}, {1,0}, 41] (* G. C. Greubel, Feb 19 2023 *)

a(n)=(2*2^n+(-4)^n)/3 \\ Charles R Greathouse IV, Oct 07 2015

[(2^(n+1)+(-4)^n)/3 for n in range(41)] # G. C. Greubel, Feb 19 2023

a(n) = (2*2^n + (-4)^n)/3.
G.f.: (1+2*x)/((1+4*x)*(1-2*x)).
E.g.f.: (2*exp(2*x) + exp(-4*x))/3.
a(n) = (-1)^n*A000079(n)*A078008(n). - Paul Barry, Feb 12 2004
a(n) = -2*a(n-1) + 8*a(n-2). - Vincenzo Librandi, Nov 12 2011

A208532 Mirror image of triangle in A125185; unsigned version of A120058.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 4, 9, 10, 4, 5, 16, 28, 24, 8, 6, 25, 60, 80, 56, 16, 7, 36, 110, 200, 216, 128, 32, 8, 49, 182, 420, 616, 560, 288, 64, 9, 64, 280, 784, 1456, 1792, 1408, 640, 128, 10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256

Offset: 0

Philippe Deléham, Feb 27 2012

Subtriangle of the triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Equals A007318*A134309*A097806 as infinite lower triangular matrix.
Row sums are powers of 3 (A000244).
Diagonal sums are powers of 2 (A000079).

			Triangle begins :
1
2, 1
3, 4, 2
4, 9, 10, 4
5, 16, 28, 24, 8
6, 25, 60, 80, 56, 16
7, 36, 110, 200, 216, 128, 32
8, 49, 182, 420, 616, 560, 288, 64
9, 64, 280, 784, 1456, 1792, 1408, 640, 128
10, 81, 408, 1344, 3024, 4704, 4992, 3456, 1408, 256
Triangle (1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 1, 0, 0, 0, ...) begins :
1
1, 0
2, 1, 0
3, 4, 2, 0
4, 9, 10, 4, 0
5, 16, 28, 24, 8, 0
6, 25, 60, 80, 56, 16, 0

Cf. Columns: A000027, A000290, A006331, A112742.

Cf. Diagonals: A011782, 2*A045623,

T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-1,k-1), T(0,0) = T(1,1) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1-y*x)/((1-x)*(1-(1+2*y)*x)).
Sum_{k, 0<=k<=n} T(n,k)*x^k = A083085(n), A084567(n), A000012(n), A000027(n+1), A000244(n), A083065(n), A083076(n) for x = -3, -2, -1, 0, 1, 2, 3 respectively.

cp's OEIS Frontend

A083086 a(n) = (2^(n+1) + (-4)^n)/3.

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