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A092686 Triangle, read by rows, such that the convolution of each row with {1,2} produces a triangle which, when flattened, equals this flattened form of the original triangle.

%I A092686 #18 Jun 13 2017 21:56:02
%S A092686 1,2,2,6,4,6,16,14,12,16,46,40,40,32,46,132,120,112,110,92,132,384,
%T A092686 352,334,312,316,264,384,1120,1038,980,940,896,912,768,1120,3278,3056,
%U A092686 2900,2776,2704,2592,2656,2240,3278,9612,9012,8576,8256,8000,7840,7552,7758
%N A092686 Triangle, read by rows, such that the convolution of each row with {1,2} produces a triangle which, when flattened, equals this flattened form of the original triangle.
%C A092686 First column and main diagonal forms A092687. Row sums form A092688.
%C A092686 This triangle is the cascadence of binomial (1+2x). More generally, the cascadence of polynomial F(x) of degree d, F(0)=1, is a triangle with d*n+1 terms in row n where the g.f. of the triangle, A(x,y), is given by: A(x,y) = ( x*H(x) - y*H(x*y^d) )/( x*F(y) - y ), where H(x) satisfies: H(x) = G*H(x*G^d)/x and G=G(x) satisfies: G(x) = x*F(G(x)) so that G = series_reversion(x/F(x)); also, H(x) is the g.f. of column 0. - _Paul D. Hanna_, Jul 17 2006
%H A092686 Paul D. Hanna, <a href="/A092686/b092686.txt">Table of n, a(n) for n = 0..495</a>
%F A092686 T(n, k) = 2*T(n-1, k) + T(n-1, k+1) for 0<=k<n, with T(n, n)=T(n, 0), T(0, 0)=1, T(0, 1)=T(1, 0)=2.
%F A092686 G.f.: A(x,y) = ( x*H(x) - y*H(x*y) )/( x*(1+2y) - y ), where H(x) satisfies: H(x) = H(x^2/(1-2x))/(1-2x) and H(x) is the g.f. of column 0 (A092687). - _Paul D. Hanna_, Jul 17 2006
%e A092686 Rows begin:
%e A092686 1;
%e A092686 2, 2;
%e A092686 6, 4, 6;
%e A092686 16, 14, 12, 16;
%e A092686 46, 40, 40, 32, 46;
%e A092686 132, 120, 112, 110, 92, 132;
%e A092686 384, 352, 334, 312, 316, 264, 384;
%e A092686 1120, 1038, 980, 940, 896, 912, 768, 1120;
%e A092686 3278, 3056, 2900, 2776, 2704, 2592, 2656, 2240, 3278;
%e A092686 9612, 9012, 8576, 8256, 8000, 7840, 7552, 7758, 6556, 9612;
%e A092686 28236, 26600, 25408, 24512, 23840, 23232, 22862, 22072, 22724, 19224, 28236; ...
%e A092686 Convolution of each row with {1,2} results in the triangle:
%e A092686 1, 2;
%e A092686 2, 6, 4;
%e A092686 6, 16, 14, 12;
%e A092686 16, 46, 40, 40, 32;
%e A092686 46, 132, 120, 112, 110, 92;
%e A092686 132, 384, 352, 334, 312, 316, 264;
%e A092686 384, 1120, 1038, 980, 940, 896, 912, 768; ...
%e A092686 which, when flattened, equals the original triangle in flattened form.
%o A092686 (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 A092686 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%o A092686 (PARI) /* Generate Triangle by the G.F.: */
%o A092686 {T(n,k)=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); polcoeff(polcoeff(A,n,x),k,y)}
%o A092686 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print()) \\ _Paul D. Hanna_, Jul 17 2006
%Y A092686 Cf. A092683, A092687, A092688, A092689, A120894, A120898.
%K A092686 nonn,tabl
%O A092686 0,2
%A A092686 _Paul D. Hanna_, Mar 04 2004