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A038220 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.

%I A038220 #33 Jul 28 2018 11:53:25
%S A038220 1,3,2,9,12,4,27,54,36,8,81,216,216,96,16,243,810,1080,720,240,32,729,
%T A038220 2916,4860,4320,2160,576,64,2187,10206,20412,22680,15120,6048,1344,
%U A038220 128,6561,34992,81648,108864,90720,48384,16128,3072,256
%N A038220 Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
%C A038220 Row sums give A000351; central terms give A119309. - _Reinhard Zumkeller_, May 14 2006
%C A038220 Triangle of coefficients in expansion of (3 + 2x)^n, where n is a nonnegative integer. - _Zagros Lalo_, Jul 23 2018
%D A038220 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48
%H A038220 Reinhard Zumkeller, <a href="/A038220/b038220.txt">Rows n = 0..125 of triangle, flattened</a>
%H A038220 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.
%H A038220 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A038220 T(n,k) = A007318(n,k) * A036561(n,k). - _Reinhard Zumkeller_, May 14 2006
%F A038220 G.f.: 1/(1 - 3*x - 2*x*y). - _Ilya Gutkovskiy_, Apr 21 2017
%F A038220 T(0,0) = 1; T(n,k) = 3 T(n-1,k) + 2 T(n-1,k-1) for k = 0...n; T(n,k)=0 for n or k < 0. - _Zagros Lalo_, Jul 23 2018
%e A038220 Triangle begins:
%e A038220    1;
%e A038220    3,   2;
%e A038220    9,  12,   4;
%e A038220   27,  54,  36,   8;
%e A038220   81, 216, 216,  96,  16;
%e A038220   ...
%t A038220 t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 3 t[n - 1, k] + 2 t[n - 1, k - 1]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Zagros Lalo_, Jul 23 2018 *)
%t A038220 Table[CoefficientList[ Expand[(3 + 2x)^n], x], {n, 0, 9}] // Flatten  (* _Zagros Lalo_, Jul 23 2018 *)
%t A038220 Table[CoefficientList[Binomial[i, j] *3^(i - j)*2^j, x], {i, 0, 9}, {j, 0, i}] // Flatten (* _Zagros Lalo_, Jul 23 2018 *)
%o A038220 (Haskell)
%o A038220 a038220 n k = a038220_tabl !! n !! k
%o A038220 a038220_row n = a038220_tabl !! n
%o A038220 a038220_tabl = iterate (\row ->
%o A038220    zipWith (+) (map (* 3) (row ++ [0])) (map (* 2) ([0] ++ row))) [1]
%o A038220 -- _Reinhard Zumkeller_, May 26 2013, Apr 02 2011
%o A038220 (PARI) T(i,j)=binomial(i,j)*3^(i-j)*2^j \\ _Charles R Greathouse IV_, Jul 19 2016
%Y A038220 Cf. A013620, A000079, A000244, A013613, A038221.
%K A038220 nonn,tabl,easy
%O A038220 0,2
%A A038220 _N. J. A. Sloane_