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%I #11 Jun 13 2017 23:48:28
%S 1,4,4,4,20,36,32,16,20,116,256,288,212,144,80,116,720,1776,2388,2144,
%T 1504,1012,784,464,720,4656,12372,18800,19632,15604,10848,7648,5712,
%U 4736,2880,4656,30996,86912,144320,169332,151792,113456,79696,58176
%N Cascadence of (1+2x)^2; a triangle, read by rows of 2n+1 terms, that retains its original form upon convolving each row with [1,4,4] and then letting excess terms spill over from each row into the initial positions of the next row such that only 2n+1 terms remain in row n for n>=0.
%C 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 satisfies: G = x*F(G) so that G = series_reversion(x/F(x)); also, H(x) is the g.f. of column 0.
%F G.f.: A(x,y) = ( x*H(x) - y*H(x*y^2) )/( x*(1+2y)^2 - y ), where H(x) satisfies: H(x) = G*H(x*G^2)/x and G satisfies: G = x*(1 + 2G)^2 ; also, H(x) is the g.f. of column 0.
%e Triangle begins:
%e 1;
%e 4, 4, 4;
%e 20, 36, 32, 16, 20;
%e 116, 256, 288, 212, 144, 80, 116;
%e 720, 1776, 2388, 2144, 1504, 1012, 784, 464, 720;
%e 4656, 12372, 18800, 19632, 15604, 10848, 7648, 5712, 4736, 2880, 4656;
%e Convolution of [1,4,4] with each row produces:
%e [1,4,4]*[1] = [1,4,4];
%e [1,4,4]*[4,4,4] = [4,20,36,32,16];
%e [1,4,4]*[20,36,32,16,20] = [20,116,256,288,212,144,80];
%e [1,4,4]*[116,256,288,212,144,80,116] =
%e [116,720,1776,2388,2144,1504,1012,784,464];
%e These convoluted rows, when concatenated, yield the sequence:
%e 1,4,4, 4,20,36,32,16, 20,116,256,288,212,144,80, 116,720,1776,2388,...
%e which equals the concatenated rows of this original triangle:
%e 1, 4,4,4, 20,36,32,16,20, 116,256,288,212,144,80,116, 720,1776,2388,...
%o (PARI) /* Generate Triangle by the Recurrence: */
%o {T(n,k)=if(2*n<k || k<0,0,if(n==0 && k==0,1,if(k==0,T(n-1,1)+4*T(n-1,0), if(k==2*n,T(n,0),T(n-1,k+1)+4*T(n-1,k)+4*T(n-1,k-1)))))}
%o for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%o (PARI) /* Generate Triangle by the G.F.: */
%o {T(n,k)=local(A,F=1+4*x+4*x^2,d=2,G=x,H=1+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 for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))
%Y Cf. A120915 (column 0), A120917 (central terms), A120918 (row sums), A000108 (Catalan); variants: A092683, A092686, A120894, A120898, A120919.
%K nonn,tabf
%O 0,2
%A _Paul D. Hanna_, Jul 17 2006
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