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A013620 Triangle of coefficients in expansion of (2+3x)^n.

%I A013620 #28 Jul 08 2025 02:48:36
%S A013620 1,2,3,4,12,9,8,36,54,27,16,96,216,216,81,32,240,720,1080,810,243,64,
%T A013620 576,2160,4320,4860,2916,729,128,1344,6048,15120,22680,20412,10206,
%U A013620 2187,256,3072,16128,48384,90720,108864,81648,34992,6561,512
%N A013620 Triangle of coefficients in expansion of (2+3x)^n.
%C A013620 Row sums give A000351; central terms give A119309. - _Reinhard Zumkeller_, May 14 2006
%H A013620 Reinhard Zumkeller, <a href="/A013620/b013620.txt">Rows n = 0..125 of triangle, flattened</a>
%H A013620 Gábor Kallós, <a href="http://dx.doi.org/10.5802/ambp.211">A generalization of Pascal’s triangle using powers of base numbers</a>, Annales mathématiques Blaise Pascal, 13 no. 1 (2006), p. 1-15.
%H A013620 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A013620 G.f.: 1 / [1 - x(2+3y)].
%F A013620 T(n,k) = A007318(n,k) * A036561(n,k). - _Reinhard Zumkeller_, May 14 2006
%e A013620 Triangle begins:
%e A013620 1;
%e A013620 2,3;
%e A013620 4,12,9;
%e A013620 8,36,54,27;
%e A013620 16,96,216,216,81;
%t A013620 Flatten[Table[Binomial[i, j] 2^(i-j) 3^j, {i, 0, 10}, {j, 0, i}]] (* _Vincenzo Librandi_, Apr 22 2014 *)
%o A013620 (Haskell)
%o A013620 a013620 n k = a013620_tabl !! n !! k
%o A013620 a013620_row n = a013620_tabl !! n
%o A013620 a013620_tabl = iterate (\row ->
%o A013620    zipWith (+) (map (* 2) (row ++ [0])) (map (* 3) ([0] ++ row))) [1]
%o A013620 -- _Reinhard Zumkeller_, May 26 2013, Apr 02 2011
%Y A013620 Cf. A038220, A000079, A000244, A013613.
%K A013620 tabl,nonn,easy
%O A013620 0,2
%A A013620 _N. J. A. Sloane_