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A189187 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).

%I A189187 #14 Apr 29 2022 03:45:25
%S A189187 1,1,1,2,3,1,4,7,5,1,7,17,16,7,1,13,38,46,29,9,1,24,82,122,99,46,11,1,
%T A189187 44,174,304,303,184,67,13,1,81,362,728,857,641,309,92,15,1,149,743,
%U A189187 1690,2291,2031,1212,482,121,17,1,274,1509,3827,5869,6004,4260,2108,711,154,19,1
%N A189187 Riordan matrix (1/(1-x-x^2-x^3),(x+x^2)/(1-x-x^2-x^3)).
%C A189187 Row sums are A077936, diagonal sums are A077946
%F A189187 T(n,k) = [x^n](x+x^2)^k/(1-x-x^2-x^3)^(k+1).
%F A189187 T(n,k) = sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j=0..n-k-2*i),i=0..n).
%F A189187 T(n,k) = sum(binomial(k,i)*(-1)^(k-i)*sum(binomial(j+k,k)*trinomial(i+j,n-3*k+2*i-j),j=0..n-k),i=0..k)
%F A189187 Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n+1,k+1) + T(n+1,k) + T(n,k+1)
%e A189187 Triangle begins:
%e A189187 1
%e A189187 1,1
%e A189187 2,3,1
%e A189187 4,7,5,1
%e A189187 7,17,16,7,1
%e A189187 13,38,46,29,9,1
%e A189187 24,82,122,99,46,11,1
%e A189187 44,174,304,303,184,67,13,1
%e A189187 81,362,728,857,641,309,92,15,1
%t A189187 Flatten[Table[Sum[Binomial[i+k,k]Sum[Binomial[i+k,j]Binomial[n-i-j,i+k],{j,0,n-k-2i}],{i,0,n}],{n,0,20},{k,0,n}]]
%o A189187 (Maxima) create_list(sum(binomial(i+k,k)*sum(binomial(i+k,j)*binomial(n-i-j,i+k),j,0,n-k-2*i),i,0,n),n,0,8,k,0,n);
%Y A189187 Cf. A104580, A187889, A077936, A077946.
%K A189187 nonn,easy,tabl
%O A189187 0,4
%A A189187 _Emanuele Munarini_, Apr 18 2011
%E A189187 a(23) and a(40) corrected by _Georg Fischer_, Feb 20 2021 and Apr 29 2022