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

%I A187889 #19 Dec 26 2023 10:08:03
%S A187889 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,
%T A187889 1,44,200,375,375,216,72,13,1,81,418,913,1100,799,359,98,15,1,149,861,
%U A187889 2147,3027,2657,1507,554,128,17,1,274,1753,4914,7937,8174,5610,2603,809,162,19,1
%N A187889 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2+x^3)/(1-x-x^2-x^3)).
%F A187889 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).
%F A187889 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)
%e A187889 Triangle begins:
%e A187889 1
%e A187889 1,1
%e A187889 2,3,1
%e A187889 4,8,5,1
%e A187889 7,19,18,7,1
%e A187889 13,43,54,32,9,1
%e A187889 24,94,147,117,50,11,1
%e A187889 44,200,375,375,216,72,13,1
%e A187889 81,418,913,1100,799,359,98,15,1
%t A187889 (* Function RiordanSquare defined in A321620. *)
%t A187889 RiordanSquare[1/(1 - x - x^2- x^3), 11] // Flatten (* _Peter Luschny_, Nov 27 2018 *)
%o A187889 (Maxima) trinomial(n,k):=coeff(expand((1+x+x^2)^n),x,k);
%o A187889 create_list(sum(binomial(i+k,k)*trinomial(i+k,n-k-i),i,0,n-k),n,0,8,k,0,n);
%Y A187889 Cf. A104580, A321620.
%K A187889 nonn,easy,tabl
%O A187889 0,4
%A A187889 _Emanuele Munarini_, Mar 15 2011