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A013611
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Triangle of coefficients in expansion of (1+4x)^n.
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7
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1, 1, 4, 1, 8, 16, 1, 12, 48, 64, 1, 16, 96, 256, 256, 1, 20, 160, 640, 1280, 1024, 1, 24, 240, 1280, 3840, 6144, 4096, 1, 28, 336, 2240, 8960, 21504, 28672, 16384, 1, 32, 448, 3584, 17920, 57344, 114688, 131072, 65536, 1, 36, 576, 5376, 32256, 129024, 344064, 589824, 589824, 262144
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OFFSET
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0,3
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COMMENTS
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T(n,k) equals the number of n-length words on {0,1,2,3,4} having n-k zeros. - Milan Janjic, Jul 24 2015
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LINKS
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J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients for k rooks on the 4xn board, all heights 4.
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FORMULA
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G.f.: 1 / (1 - x(1+4y)).
T(n,k) = 4^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*3^(n-i). Row sums are 5^n = A000351. - Mircea Merca, Apr 28 2012
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EXAMPLE
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Triangle begins
1;
1, 4;
1, 8, 16;
1, 12, 48, 64;
1, 16, 96, 256, 256;
1, 20, 160, 640, 1280, 1024;
1, 24, 240, 1280, 3840, 6144, 4096;
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MAPLE
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T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+4*x)^n):
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MATHEMATICA
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Flatten[Table[CoefficientList[Series[(1+4x)^n, {x, 0, 10}], x], {n, 0, 15}]] (* Harvey P. Dale, Oct 10 2011 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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