A218016 Triangle, read by rows, where T(n,k) = k!*C(n, k)*5^(n-k) for n>=0, k=0..n.

1, 5, 1, 25, 10, 2, 125, 75, 30, 6, 625, 500, 300, 120, 24, 3125, 3125, 2500, 1500, 600, 120, 15625, 18750, 18750, 15000, 9000, 3600, 720, 78125, 109375, 131250, 131250, 105000, 63000, 25200, 5040, 390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 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=5.

Sum(T(n,k), k=0..n) = A080954(n) (see the Formula section of A080954). . Also:

first column: A000351;

second column: A053464;

third column: 2*A084902;

fourth column: 6*A081143.

Examples

			Triangle begins:
1;
5,      1;
25,     10,     2;
125,    75,     30,     6;
625,    500,    300,    120,     24;
3125,   3125,   2500,   1500,    600,     120;
15625,  18750,  18750,  15000,   9000,    3600,   720;
78125,  109375, 131250, 131250,  105000,  63000,  25200,  5040;
390625, 625000, 875000, 1050000, 1050000, 840000, 504000, 201600, 40320; etc.
		

Links

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

Crossrefs

Cf. A000351, A053464, A080954, A081143, A084902, A090802, A217629, A218017.

Programs

Magma

[Factorial(n)/Factorial(n-k)*5^(n-k): k in [0..n], n in [0..10]];

Mathematica

Flatten[Table[n!/(n-k)!*5^(n-k), {n, 0, 10}, {k, 0, n}]]

Formula

T(n,k) = 5^(n-k)*n!/(n-k)! for n>=0, k=0..n.

E.g.f. (by columns): exp(5x)*x^k.