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Cascadence of (1+x)^3; a triangle, read by rows of 3n+1 terms, that retains its original form upon convolving each row with [1,3,3,1] and then letting excess terms spill over from each row into the initial positions of the next row such that only 3n+1 terms remain in row n for n>=0.
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%I #11 Jan 24 2018 03:19:05

%S 1,3,3,1,3,12,19,18,15,10,3,12,55,111,138,128,96,66,55,39,12,55,276,

%T 636,930,1005,876,669,498,360,263,240,177,55,276,1464,3666,5979,7317,

%U 7242,6138,4737,3506,2607,2046,1569,1212,1170,883,276,1464,8058,21369,37716

%N Cascadence of (1+x)^3; a triangle, read by rows of 3n+1 terms, that retains its original form upon convolving each row with [1,3,3,1] and then letting excess terms spill over from each row into the initial positions of the next row such that only 3n+1 terms remain in row n for n>=0.

%F G.f.: A(x,y) = ( x*H(x) - y*H(x*y^3) )/( x*(1+y)^3 - y ), where H(x) satisfies: H(x) = G*H(x^4*G^3) and G(x) is g.f. of A001764: G(x) = 1 + x*G(x)^3.

%e Triangle begins:

%e 1;

%e 3, 3, 1, 3;

%e 12, 19, 18, 15, 10, 3, 12;

%e 55, 111, 138, 128, 96, 66, 55, 39, 12, 55;

%e 276, 636, 930, 1005, 876, 669, 498, 360, 263, 240, 177, 55, 276;

%e Convolution of [1,3,3,1] with each row produces:

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

%e [1,3,3,1]*[3,3,1,3] = [3,12,19,18,15,10,3];

%e [1,3,3,1]*[12,19,18,15,10,3,12] = [12,55,111,138,128,96,66,55,39,12];

%e [1,3,3,1]*[55,111,138,128,96,66,55,39,12,55] =

%e [55,276,636,930,1005,876,669,498,360,263,240,177,55];

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

%e 1,3,3,1, 3,12,19,18,15,10,3, 12,55,111,138,128,96,66,55,39,12, 55,...

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

%e 1, 3,3,1,3, 12,19,18,15,10,3,12, 55,111,138,128,96,66,55,39,12,55, ...

%t T[n_, k_] := T[n, k] = If[3*n < k || k < 0, 0, If[n == 0 && k == 0, 1, If[k == 3*n, T[n, 0], T[n - 1, k + 1] + 3*T[n - 1, k] + 3*T[n - 1, k - 1] + T[n - 1, k - 2]]]];

%t Table[T[n, k], {n, 0, 10}, {k, 0, 3 n}] // Flatten (* _Jean-François Alcover_, Jan 24 2018, translated from PARI *)

%o (PARI) /* Generate Triangle by the Recurrence: */

%o {T(n,k)=if(3*n<k || k<0,0,if(n==0 && k==0,1,if(k==3*n,T(n,0), T(n-1,k+1)+3*T(n-1,k)+3*T(n-1,k-1)+T(n-1,k-2))))}

%o for(n=0, 10, for(k=0, 3*n, print1(T(n, k), ", ")); print(""))

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

%o {T(n,k)=local(A,F=(1+x)^3,d=3,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, 3*n, print1(T(n, k), ", ")); print(""))

%Y Cf. A120920 (column 0), A120922 (central terms), A120923 (row sums), A001764 (ternary trees); variants: A092683, A092686, A120894, A120898, A120919.

%K nonn,tabl

%O 0,2

%A _Paul D. Hanna_, Jul 17 2006