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A013611 Triangle of coefficients in expansion of (1+4x)^n.

%I A013611 #48 Apr 23 2019 23:57:13
%S A013611 1,1,4,1,8,16,1,12,48,64,1,16,96,256,256,1,20,160,640,1280,1024,1,24,
%T A013611 240,1280,3840,6144,4096,1,28,336,2240,8960,21504,28672,16384,1,32,
%U A013611 448,3584,17920,57344,114688,131072,65536,1,36,576,5376,32256,129024,344064,589824,589824,262144
%N A013611 Triangle of coefficients in expansion of (1+4x)^n.
%C A013611 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
%H A013611 Harvey P. Dale, <a href="/A013611/b013611.txt">Table of n, a(n) for n = 0..1034</a> (rows 0..44 flattened, missing terms added by Sean A. Irvine, Apr 21 2019)
%H A013611 J. Goldman, J. Haglund, <a href="http://dx.doi.org/10.1006/jcta.2000.3113">Generalized rook polynomials</a>, J. Combin. Theory A91 (2000), 509-530, 1-rook coefficients for k rooks on the 4xn board, all heights 4.
%F A013611 G.f.: 1 / (1 - x(1+4y)).
%F A013611 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
%e A013611 Triangle begins
%e A013611   1;
%e A013611   1,    4;
%e A013611   1,    8,   16;
%e A013611   1,   12,   48,   64;
%e A013611   1,   16,   96,  256,  256;
%e A013611   1,   20,  160,  640, 1280, 1024;
%e A013611   1,   24,  240, 1280, 3840, 6144, 4096;
%p A013611 T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+4*x)^n):
%p A013611 seq(T(n), n=0..10);  # _Alois P. Heinz_, Jul 24 2015
%t A013611 Flatten[Table[CoefficientList[Series[(1+4x)^n,{x,0,10}],x],{n,0,15}]] (* _Harvey P. Dale_, Oct 10 2011 *)
%Y A013611 Cf. A000351 (5^n).
%K A013611 tabl,nonn,easy
%O A013611 0,3
%A A013611 _N. J. A. Sloane_