%I #20 Jun 13 2017 21:52:31
%S 1,1,1,2,1,2,3,3,2,3,6,5,5,3,6,11,10,8,9,6,11,21,18,17,15,17,11,21,39,
%T 35,32,32,28,32,21,39,74,67,64,60,60,53,60,39,74,141,131,124,120,113,
%U 113,99,113,74,141,272,255,244,233,226,212,212,187,215,141,272,527,499
%N Triangle, read by rows, such that the convolution of each row with {1,1} produces a triangle which, when flattened, equals this flattened form of the original triangle.
%C First column and main diagonal forms A092684. Row sums form A092685.
%C This triangle is the cascadence of binomial (1+x). 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 Paul D. Hanna, <a href="/A092683/b092683.txt">Table of n, a(n) for n = 0..1325; rows 0..50 in flattened form.</a>
%F T(n, k) = 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)=1.
%F G.f.: A(x,y) = ( x*H(x) - y*H(x*y) )/( x*(1+y) - y ), where H(x) satisfies: H(x) = H(x^2/(1-x))/(1-x) and H(x) is the g.f. of column 0 (A092684). - _Paul D. Hanna_, Jul 17 2006
%e Rows begin:
%e 1;
%e 1, 1;
%e 2, 1, 2;
%e 3, 3, 2, 3;
%e 6, 5, 5, 3, 6;
%e 11, 10, 8, 9, 6, 11;
%e 21, 18, 17, 15, 17, 11, 21;
%e 39, 35, 32, 32, 28, 32, 21, 39;
%e 74, 67, 64, 60, 60, 53, 60, 39, 74;
%e 141, 131, 124, 120, 113, 113, 99, 113, 74, 141;
%e 272, 255, 244, 233, 226, 212, 212, 187, 215, 141, 272;
%e 527, 499, 477, 459, 438, 424, 399, 402, 356, 413, 272, 527;
%e 1026, 976, 936, 897, 862, 823, 801, 758, 769, 685, 799, 527, 1026; ...
%e The convolution of each row with {1,1} gives the triangle:
%e 1, 1;
%e 1, 2, 1;
%e 2, 3, 3, 2;
%e 3, 6, 5, 5, 3;
%e 6, 11, 10, 8, 9, 6;
%e 11, 21, 18, 17, 15, 17, 11;
%e 21, 39, 35, 32, 32, 28, 32, 21;
%e 39, 74, 67, 64, 60, 60, 53, 60, 39; ...
%e which, when flattened, equals the original triangle in flattened form.
%o (PARI) T(n,k)=if(n<0 || k>n,0, if(n==0 && k==0,1, if(n==1 && k<=1,1, if(k==n,T(n,0), T(n-1,k)+T(n-1,k+1)))))
%o for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%o (PARI) /* Generate Triangle by G.F. where F=1+x: */
%o {T(n,k)=local(A,F=1+x,d=1,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()) \\ _Paul D. Hanna_, Jul 17 2006
%Y Cf. A092684, A092685, A092686, A092689, A120894, A120898.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Mar 04 2004
|