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

1, 3, 2, 9, 12, 4, 27, 54, 36, 8, 81, 216, 216, 96, 16, 243, 810, 1080, 720, 240, 32, 729, 2916, 4860, 4320, 2160, 576, 64, 2187, 10206, 20412, 22680, 15120, 6048, 1344, 128, 6561, 34992, 81648, 108864, 90720, 48384, 16128, 3072, 256

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• tabl
• easy

Comments

Row sums give A000351; central terms give A119309. - Reinhard Zumkeller, May 14 2006

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

Examples

			Triangle begins:
   1;
   3,   2;
   9,  12,   4;
  27,  54,  36,   8;
  81, 216, 216,  96,  16;
  ...
		

References

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

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

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

Programs

Haskell

a038220 n k = a038220_tabl !! n !! k
a038220_row n = a038220_tabl !! n
a038220_tabl = iterate (\row ->
   zipWith (+) (map (* 3) (row ++ [0])) (map (* 2) ([0] ++ row))) [1]
-- Reinhard Zumkeller, May 26 2013, Apr 02 2011

Mathematica

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 *)
Table[CoefficientList[ Expand[(3 + 2x)^n], x], {n, 0, 9}] // Flatten  (* Zagros Lalo, Jul 23 2018 *)
Table[CoefficientList[Binomial[i, j] *3^(i - j)*2^j, x], {i, 0, 9}, {j, 0, i}] // Flatten (* Zagros Lalo, Jul 23 2018 *)

PARI

T(i,j)=binomial(i,j)*3^(i-j)*2^j \\ Charles R Greathouse IV, Jul 19 2016

Formula

T(n,k) = A007318(n,k) * A036561(n,k). - Reinhard Zumkeller, May 14 2006

G.f.: 1/(1 - 3*x - 2*x*y). - Ilya Gutkovskiy, Apr 21 2017

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