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Triangle whose (i,j)-th entry is binomial(i,j)*3^(i-j)*2^j.
8

%I #33 Jul 28 2018 11:53:25

%S 1,3,2,9,12,4,27,54,36,8,81,216,216,96,16,243,810,1080,720,240,32,729,

%T 2916,4860,4320,2160,576,64,2187,10206,20412,22680,15120,6048,1344,

%U 128,6561,34992,81648,108864,90720,48384,16128,3072,256

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

%C Row sums give A000351; central terms give A119309. - _Reinhard Zumkeller_, May 14 2006

%C Triangle of coefficients in expansion of (3 + 2x)^n, where n is a nonnegative integer. - _Zagros Lalo_, Jul 23 2018

%D Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 44, 48

%H Reinhard Zumkeller, <a href="/A038220/b038220.txt">Rows n = 0..125 of triangle, flattened</a>

%H 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 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F T(n,k) = A007318(n,k) * A036561(n,k). - _Reinhard Zumkeller_, May 14 2006

%F G.f.: 1/(1 - 3*x - 2*x*y). - _Ilya Gutkovskiy_, Apr 21 2017

%F 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 Triangle begins:

%e 1;

%e 3, 2;

%e 9, 12, 4;

%e 27, 54, 36, 8;

%e 81, 216, 216, 96, 16;

%e ...

%t 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 Table[CoefficientList[ Expand[(3 + 2x)^n], x], {n, 0, 9}] // Flatten (* _Zagros Lalo_, Jul 23 2018 *)

%t Table[CoefficientList[Binomial[i, j] *3^(i - j)*2^j, x], {i, 0, 9}, {j, 0, i}] // Flatten (* _Zagros Lalo_, Jul 23 2018 *)

%o (Haskell)

%o a038220 n k = a038220_tabl !! n !! k

%o a038220_row n = a038220_tabl !! n

%o a038220_tabl = iterate (\row ->

%o zipWith (+) (map (* 3) (row ++ [0])) (map (* 2) ([0] ++ row))) [1]

%o -- _Reinhard Zumkeller_, May 26 2013, Apr 02 2011

%o (PARI) T(i,j)=binomial(i,j)*3^(i-j)*2^j \\ _Charles R Greathouse IV_, Jul 19 2016

%Y Cf. A013620, A000079, A000244, A013613, A038221.

%K nonn,tabl,easy

%O 0,2

%A _N. J. A. Sloane_