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A038231 Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).

%I A038231 #40 Oct 09 2022 04:59:40
%S A038231 1,4,1,16,8,1,64,48,12,1,256,256,96,16,1,1024,1280,640,160,20,1,4096,
%T A038231 6144,3840,1280,240,24,1,16384,28672,21504,8960,2240,336,28,1,65536,
%U A038231 131072,114688,57344,17920,3584,448,32,1,262144,589824,589824,344064,129024,32256,5376,576,36,1
%N A038231 Triangle whose (i,j)-th entry is binomial(i,j)*4^(i-j).
%C A038231 Triangle of coefficients in expansion of (4+x)^n. - _N-E. Fahssi_, Apr 13 2008
%H A038231 Indranil Ghosh, <a href="/A038231/b038231.txt">Rows 0..125 of triangle, flattened</a>
%H A038231 Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
%H A038231 B. N. Cyvin et al., <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match34/match34_109-121.pdf">Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons</a>, Match, No. 34 (Oct 1996), pp. 109-121.
%F A038231 G.f. for j-th column is (x^j)/(1-4*x)^(j+1).
%F A038231 Convolution triangle of A000302 (powers of 4).
%F A038231 Sum_{k=0..n} T(n,k)*(-1)^k*A000108(k) = A001700(n). - _Philippe Deléham_, Nov 27 2009
%F A038231 See A038207 and A027465 and replace 2 and 3 in analogous formulas with 4. - _Tom Copeland_, Oct 26 2012
%e A038231 Triangle begins:
%e A038231       1;
%e A038231       4,      1;
%e A038231      16,      8,      1;
%e A038231      64,     48,     12,     1;
%e A038231     256,    256,     96,    16,     1;
%e A038231    1024,   1280,    640,   160,    20,    1;
%e A038231    4096,   6144,   3840,  1280,   240,   24,   1;
%e A038231   16384,  28672,  21504,  8960,  2240,  336,  28,  1;
%e A038231   65536, 131072, 114688, 57344, 17920, 3584, 448, 32, 1;
%p A038231 for i from 0 to 10 do seq(binomial(i, j)*4^(i-j), j = 0 .. i) od; # _Zerinvary Lajos_, Dec 21 2007
%p A038231 # Uses function PMatrix from A357368. Adds column 1, 0, 0, ... to the left.
%p A038231 PMatrix(10, n -> 4^(n-1)); # _Peter Luschny_, Oct 09 2022
%t A038231 Table[4^(n-k)*Binomial[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 20 2019 *)
%o A038231 (PARI) T(n,k) = 4^(n-k)*binomial(n, k); \\ _G. C. Greubel_, Jul 20 2019
%o A038231 (Magma) [4^(n-k)*Binomial(n, k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jul 20 2019
%o A038231 (Sage) [[4^(n-k)*binomial(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Jul 20 2019
%o A038231 (GAP) Flat(List([0..10], n-> List([0..n], k-> 4^(n-k)*Binomial(n, k) ))); # _G. C. Greubel_, Jul 20 2019
%Y A038231 Cf. A000302, A013611 (row-reversed), A000351 (row sums).
%K A038231 nonn,tabl,easy
%O A038231 0,2
%A A038231 _N. J. A. Sloane_