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A triangle formed like Pascal's triangle, but with 4^n on the borders instead of 1.
5

%I #7 Aug 07 2013 00:35:24

%S 1,4,4,16,8,16,64,24,24,64,256,88,48,88,256,1024,344,136,136,344,1024,

%T 4096,1368,480,272,480,1368,4096,16384,5464,1848,752,752,1848,5464,

%U 16384,65536,21848,7312,2600,1504,2600,7312,21848,65536,262144,87384,29160

%N A triangle formed like Pascal's triangle, but with 4^n on the borders instead of 1.

%C All rows except the zeroth are divisible by 4. Is there a closed-form formula for these numbers, like for binomial coefficients?

%H T. D. Noe, <a href="/A227074/b227074.txt">Rows n = 0..50 of triangle, flattened</a>

%e Example:

%e 1,

%e 4, 4,

%e 16, 8, 16,

%e 64, 24, 24, 64,

%e 256, 88, 48, 88, 256,

%e 1024, 344, 136, 136, 344, 1024,

%e 4096, 1368, 480, 272, 480, 1368, 4096,

%e 16384, 5464, 1848, 752, 752, 1848, 5464, 16384,

%e 65536, 21848, 7312, 2600, 1504, 2600, 7312, 21848, 65536

%t t = {}; Do[r = {}; Do[If[k == 0 || k == n, m = 4^n, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]

%Y Cf. A007318 (Pascal's triangle), A228053 ((-1)^n on the borders).

%Y Cf. A051601 (n on the borders), A137688 (2^n on borders).

%Y Cf. A165665 (row sums: 3*4^n - 2*2^n), A227075 (3^n edges), A227076 (5^n edges).

%K nonn,tabl

%O 0,2

%A _T. D. Noe_, Aug 06 2013