A191239 Triangle T(n,k) = coefficient of x^n in expansion of (x+x^2+2*x^3)^k.

1, 1, 1, 2, 2, 1, 0, 5, 3, 1, 0, 4, 9, 4, 1, 0, 4, 13, 14, 5, 1, 0, 0, 18, 28, 20, 6, 1, 0, 0, 12, 49, 50, 27, 7, 1, 0, 0, 8, 56, 105, 80, 35, 8, 1, 0, 0, 0, 56, 161, 195, 119, 44, 9, 1, 0, 0, 0, 32, 210, 366, 329, 168, 54, 10, 1, 0, 0, 0, 16, 200, 581, 721, 518, 228, 65, 11, 1, 0, 0, 0, 0, 160, 732, 1337, 1288, 774, 300, 77, 12, 1, 0, 0, 0, 0, 80, 780, 2045, 2716, 2142, 1110, 385, 90, 13, 1

Offset: 1

Author

Vladimir Kruchinin, May 27 2011

Keywords

• nonn
• tabl

Comments

1. Riordan Array (1,x+x^2+2*x^3) without first column.

2. Riordan Array (1+x+2*x^3,x+x^2+2*x^3) numbering triangle (0,0).

3. Bell Polynomial of second kind B(n,k){1,2,12,0,0,0,...,0}=n!/k!*T(n,k).

4. For the g.f. 1/(1-x-x^2-2*x^3) we have a(n)=sum(k=1..n, T(n,k)) (see A077947)

For more formulas see preprints.

Examples

			Triangle begins:
  1,
  1,1,
  2,2,1,
  0,5,3,1,
  0,4,9,4,1,
  0,4,13,14,5,1,
  0,0,18,28,20,6,1,
		

Links

• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
• Vladimir Kruchinin, Derivation of Bell Polynomials of the Second Kind, arXiv:1104.5065 [math.CO], 2013.

Programs

Maxima

T(n,k):=sum(binomial(j,n-3*k+2*j)*2^(k-j)*binomial(k,j),j,0,k);

Formula

T(n,k) = Sum_{j=0..k} binomial(j,n-3*k+2*j)*2^(k-j)*binomial(k,j).