A217629 Triangle, read by rows, where T(n,k) = k!*C(n, k)*3^(n-k) for n>=0, k=0..n.

1, 3, 1, 9, 6, 2, 27, 27, 18, 6, 81, 108, 108, 72, 24, 243, 405, 540, 540, 360, 120, 729, 1458, 2430, 3240, 3240, 2160, 720, 2187, 5103, 10206, 17010, 22680, 22680, 15120, 5040, 6561, 17496, 40824, 81648, 136080, 181440, 181440, 120960, 40320

Offset: 0

Author

Vincenzo Librandi, Nov 10 2012

Keywords

• nonn
• tabl
• easy

Comments

Triangle formed by the derivatives of x^n evaluated at x=3.

Sum(T(n,k), k=0..n) = A053486(n) (see the Formula section of A053486). Also:

first column: A000244;

second column: A027471;

third column: 2*A027472;

fourth column: 6*A036216;

fifth column: 24*A036217.

Examples

			Triangle begins:
1;
3,     1;
9,     6,     2;
27,    27,    18,     6;
81,    108,   108,    72,     24;
243,   405,   540,    540,    360,    120;
729,   1458,  2430,   3240,   3240,   2160,    720;
2187,  5103,  10206,  17010,  22680,  22680,   15120,   5040;
6561,  17496, 40824,  81648,  136080, 181440,  181440,  120960,  40320; etc.
		

Links

• Vincenzo Librandi, Rows n = 0..100, flattened

Crossrefs

Cf. A000244, A027471, A027472, A036216, A036217, A053486, A090802, A218016, A218017.

Programs

Magma

[Factorial(n)/Factorial(n-k)*3^(n-k): k in [0..n], n in [0..10]];

Mathematica

Flatten[Table[n!/(n-k)!*3^(n-k), {n, 0, 10}, {k, 0, n}]]

Formula

T(n,k) = 3^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(3x)*x^k.