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A092688 Row sums of triangle A092686, in which the convolution of each row with {1,2} produces a triangle that, when flattened, equals the flattened form of A092686.

%I A092688 #7 Jun 13 2017 21:56:53
%S A092688 1,4,16,58,204,698,2346,7774,25480,82774,266946,855674,2728702,
%T A092688 8663402,27400862,86376186,271488444,851099874,2661967502,8308462182,
%U A092688 25883429326,80497346294,249956869434,775048966478,2400067860090
%N A092688 Row sums of triangle A092686, in which the convolution of each row with {1,2} produces a triangle that, when flattened, equals the flattened form of A092686.
%F A092688 G.f.: A(x) = H(x)*(1-x)/(1-3*x), where H(x) satisfies: H(x) = H(x^2/(1-2x))/(1-2x) and H(x) is the g.f. of A092687. - _Paul D. Hanna_, Jul 17 2006
%o A092688 (PARI) {T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,2, if(k==n,T(n,0), 2*T(n-1,k)+T(n-1,k+1)))))}
%o A092688 a(n)=sum(k=0,n,T(n,k))
%o A092688 (PARI) {a(n)=local(A,F=1+2*x,d=1,G=x,H=1+2*x,S=ceil(log(n+1)/log(d+1))); for(i=0,n,G=x*subst(F,x,G+x*O(x^n)));for(i=0,S,H=subst(H,x,x*G^d+x*O(x^n))*G/x); A=(x*H-y*subst(H,x,x*y^d +x*O(x^n)))/(x*subst(F,x,y)-y); sum(k=0,d*n,polcoeff(polcoeff(A,n,x),k,y))} \\ _Paul D. Hanna_, Jul 17 2006
%Y A092688 Cf. A092683, A092686, A092687, A092689.
%K A092688 nonn
%O A092688 0,2
%A A092688 _Paul D. Hanna_, Mar 04 2004