A120914 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.

1, 4, 4, 4, 20, 36, 32, 16, 20, 116, 256, 288, 212, 144, 80, 116, 720, 1776, 2388, 2144, 1504, 1012, 784, 464, 720, 4656, 12372, 18800, 19632, 15604, 10848, 7648, 5712, 4736, 2880, 4656, 30996, 86912, 144320, 169332, 151792, 113456, 79696, 58176

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..43.

Formula

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.

Example

Triangle begins:
1;
4, 4, 4;
20, 36, 32, 16, 20;
116, 256, 288, 212, 144, 80, 116;
720, 1776, 2388, 2144, 1504, 1012, 784, 464, 720;
4656, 12372, 18800, 19632, 15604, 10848, 7648, 5712, 4736, 2880, 4656;
Convolution of [1,4,4] with each row produces:
[1,4,4]*[1] = [1,4,4];
[1,4,4]*[4,4,4] = [4,20,36,32,16];
[1,4,4]*[20,36,32,16,20] = [20,116,256,288,212,144,80];
[1,4,4]*[116,256,288,212,144,80,116] =
[116,720,1776,2388,2144,1504,1012,784,464];
These convoluted rows, when concatenated, yield the sequence:
1,4,4, 4,20,36,32,16, 20,116,256,288,212,144,80, 116,720,1776,2388,...
which equals the concatenated rows of this original triangle:
1, 4,4,4, 20,36,32,16,20, 116,256,288,212,144,80,116, 720,1776,2388,...

Prog

(PARI) /* Generate Triangle by the Recurrence: */
{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)))))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* Generate Triangle by the G.F.: */
{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)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A120915 (column 0), A120917 (central terms), A120918 (row sums), A000108 (Catalan); variants: A092683, A092686, A120894, A120898, A120919.

Sequence in context: A034896 A320970 A216871 * A303397 A024949 A362374

Adjacent sequences: A120911 A120912 A120913 * A120915 A120916 A120917

Keyword

nonn,tabf

Author

Paul D. Hanna, Jul 17 2006

Status

approved