A120894 - OEIS

%I #14 Jun 13 2017 23:47:36

%S 1,1,1,1,2,3,2,1,2,5,7,6,5,3,2,5,12,18,18,14,10,10,7,5,12,30,48,50,42,

%T 34,27,22,24,17,12,30,78,128,140,126,103,83,73,63,53,59,42,30,78,206,

%U 346,394,369,312,259,219,189,175,154,131,150,108,78,206,552,946,1109

%N Cascadence of 1+x+x^2; a triangle, read by rows of 2n+1 terms, that retains its original form upon convolving each row with [1,1,1] 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 In this case, the g.f. of column 0, H(x), satisfies: H(x) = H(x*G^2)*G/x where G satisfies: G = x*(1+G+G^2), so that G/x = g.f. of Motzkin numbers (A001006). 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. H(x) of column 0 satisfies: H(x) = H(x*G^d)*G/x where G = x*F(G); thus G = series_reversion(x/F(x)), or, equivalently, [x^n] G = [x^n] x*F(x)^n/n for n>=1.

%C Further, the g.f. of the cascadence triangle for polynomial F(x) of degree d is given by: A(x,y) = ( x*H(x) - y*H(x*y^d) )/( x*F(y) - y ), where H(x) = G*H(x*G^d)/x and G = x*F(G). - _Paul D. Hanna_, Jul 17 2006

%H Paul D. Hanna, <a href="/A120894/b120894.txt">Table of n, a(n) for n = 0..440</a>

%F G.f.: A(x,y) = ( x*H(x) - y*H(x*y^2) )/( x*F(y) - y ), where H(x) = G*H(x*G^2)/x, G = x*F(G), F(x)=1+x+x^2. - _Paul D. Hanna_, Jul 17 2006

%e Triangle begins:

%e 1;

%e 1, 1, 1;

%e 2, 3, 2, 1, 2;

%e 5, 7, 6, 5, 3, 2, 5;

%e 12, 18, 18, 14, 10, 10, 7, 5, 12;

%e 30, 48, 50, 42, 34, 27, 22, 24, 17, 12, 30;

%e 78, 128, 140, 126, 103, 83, 73, 63, 53, 59, 42, 30, 78;

%e 206, 346, 394, 369, 312, 259, 219, 189, 175, 154, 131, 150, 108, 78, 206;

%e 552, 946, 1109, 1075, 940, 790, 667, 583, 518, 460, 435, 389, 336, 392, 284, 206, 552;

%e 1498, 2607, 3130, 3124, 2805, 2397, 2040, 1768, 1561, 1413, 1284, 1160, 1117, 1012, 882, 1042, 758, 552, 1498; ...

%e Convolution of [1,1,1] with each row produces:

%e [1,1,1]*[1] = [1,1,1];

%e [1,1,1]*[1,1,1] = [1,2,3,2,1];

%e [1,1,1]*[2,3,2,1,2] = [2,5,7,6,5,3,2];

%e [1,1,1]*[5,7,6,5,3,2,5] = [5,12,18,18,14,10,10,7,5];

%e [1,1,1]*[12,18,18,14,10,10,7,5,12] = [12,30,48,50,42,34,27,22,24,17,12]; ...

%e These convoluted rows, when concatenated, yield the sequence:

%e 1,1,1, 1,2,3,2,1, 2,5,7,6,5,3,2, 5,12,18,18,14,10,10,7,5, ...

%e which equals the concatenated rows of this original triangle:

%e 1, 1,1,1, 2,3,2,1,2, 5,7,6,5,3,2,5, 12,18,18,14,10,10,7,5,12, ...

%o (PARI) T(n,k)=if(2*n<k || k<0,0,if(n<=1,1,if(k==0,T(n-1,0)+T(n-1,1), if(k==2*n,T(n,0), T(n-1,k-1)+T(n-1,k)+T(n-1,k+1)))))

%o for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))

%o (PARI) /* Generated by the G.F.: */

%o {T(n,k)=local(A,F=1+x+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, 2*n, print1(T(n, k), ", ")); print()) \\ _Paul D. Hanna_, Jul 17 2006

%Y Cf. A120895 (column 0), A120896 (central terms), A120897 (row sums), A001006 (Motzkin numbers); variants: A092683, A092686, A120898.

%K nonn,tabf

%O 0,5

%A _Paul D. Hanna_, Jul 14 2006