A187889 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2+x^3)/(1-x-x^2-x^3)).

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 7, 19, 18, 7, 1, 13, 43, 54, 32, 9, 1, 24, 94, 147, 117, 50, 11, 1, 44, 200, 375, 375, 216, 72, 13, 1, 81, 418, 913, 1100, 799, 359, 98, 15, 1, 149, 861, 2147, 3027, 2657, 1507, 554, 128, 17, 1, 274, 1753, 4914, 7937, 8174, 5610, 2603, 809, 162, 19, 1

Offset: 0

Author

Emanuele Munarini, Mar 15 2011

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins:
1
1,1
2,3,1
4,8,5,1
7,19,18,7,1
13,43,54,32,9,1
24,94,147,117,50,11,1
44,200,375,375,216,72,13,1
81,418,913,1100,799,359,98,15,1
		

Crossrefs

Cf. A104580, A321620.

Programs

Mathematica

(* Function RiordanSquare defined in A321620. *)
RiordanSquare[1/(1 - x - x^2- x^3), 11] // Flatten (* Peter Luschny, Nov 27 2018 *)

Maxima

trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
create_list(sum(binomial(i+k,k)*trinomial(i+k,n-k-i),i,0,n-k),n,0,8,k,0,n);

Formula

a(n,k) = Sum_{i=0..n-k} binomial(i+k,k)*trinomial(i+k,n-k-i), where trinomial(n,k) are the trinomial coefficients (A027907).

Recurrence: a(n+3,k+1) = a(n+2,k+1) + a(n+2,k) + a(n+1,k+1) + a(n+1,k) + a(n,k+1) + a(n,k)