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

0,4

Links

Table of n, a(n) for n=0..65.

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)

Example

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

Mathematica

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

Prog

(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);

Crossrefs

Cf. A104580, A321620.

Sequence in context: A021436 A179738 A377307 * A353593 A118800 A200139

Adjacent sequences: A187886 A187887 A187888 * A187890 A187891 A187892

Keyword

nonn,easy,tabl

Author

Emanuele Munarini, Mar 15 2011

Status

approved