A099093 Riordan array (1, 3+3x).

1, 0, 3, 0, 3, 9, 0, 0, 18, 27, 0, 0, 9, 81, 81, 0, 0, 0, 81, 324, 243, 0, 0, 0, 27, 486, 1215, 729, 0, 0, 0, 0, 324, 2430, 4374, 2187, 0, 0, 0, 0, 81, 2430, 10935, 15309, 6561, 0, 0, 0, 0, 0, 1215, 14580, 45927, 52488, 19683, 0, 0, 0, 0, 0, 243, 10935, 76545, 183708, 177147, 59049

Offset: 0

Author

Paul Barry, Sep 25 2004

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A030195. Diagonal sums are A099094.

The Riordan array (1,s+tx) defines T(n,k) = binomial(k,n-k)s^k(t/s)^(n-k). The row sums satisfy a(n)=s*a(n-1)+t*a(n-2) and the diagonal sums satisfy a(n)=s*a(n-2)+t*a(n-3).

Modulo 2, this sequence gives A106344. - Philippe Deléham, Dec 18 2008

Examples

			Rows begin:
  1;
  0, 3;
  0, 3, 9;
  0, 0, 18, 27;
  0, 0, 9, 81, 81;
  0, 0, 0, 81, 324, 243;
  0, 0, 0, 27, 486, 1215, 729;
  ...
		

Crossrefs

Cf. A038221.

Programs

Magma

[[Binomial(k,n-k)*3^k: k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Feb 21 2015 /* as the triangle */

PARI

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(k, n-k)*3^k, ", ");); print(););} \\ Michel Marcus, Feb 21 2015

Formula

T(n,k) = binomial(k, n-k)*3^k. - corrected by Michel Marcus, Feb 21 2015

Columns have g.f. (3x+3x^3)^k.

T(n,k) = A026729(n,k)*3^k. - Philippe Deléham, Jul 29 2006