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 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified September 16 12:35 EDT 2021. Contains 347472 sequences. (Running on oeis4.)