A166884 Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x+x^2+x^3 (cf. A166880).
1, 1, 1, 3, 2, 1, 15, 9, 3, 1, 114, 62, 18, 4, 1, 1159, 593, 157, 30, 5, 1, 14838, 7266, 1812, 316, 45, 6, 1, 229401, 108720, 25989, 4271, 555, 63, 7, 1, 4159662, 1922166, 445255, 70180, 8595, 890, 84, 8, 1, 86580636, 39212154, 8865333, 1354750, 159171, 15534, 1337, 108, 9, 1
Offset: 0
Examples
This triangle begins: 1; 1, 1; 3, 2, 1; 15, 9, 3, 1; 114, 62, 18, 4, 1; 1159, 593, 157, 30, 5, 1; 14838, 7266, 1812, 316, 45, 6, 1; 229401, 108720, 25989, 4271, 555, 63, 7, 1; 4159662, 1922166, 445255, 70180, 8595, 890, 84, 8, 1; 86580636, 39212154, 8865333, 1354750, 159171, 15534, 1337, 108, 9, 1; 2034850425, 906623004, 201058614, 30000676, 3418245, 320070, 25963, 1912, 135, 10, 1; 53303009286, 23429034168, 5114874693, 748896765, 83336385, 7568355, 589057, 40882, 2631, 165, 11, 1; ... Triangle A166880 of coefficients in iterations of x+x^2+x^3 begins: 1; 1,1,1; 1,2,4,6,8,8,6,3,1; 1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,...; 1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...; 1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...; 1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...; 1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...; 1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...; ... in which this triangle transforms diagonals in A166880 into each other. The initial diagonals in triangle A166880 begin: A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...]; A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...]; A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ... so that, if we treat the diagonals as column vectors, we have: A166884 * A166881 = A166882, A166884 * A166882 = A166883.
Links
- Paul D. Hanna, a(n) for n = 0..350 (rows 0..25, flattened).
Programs
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PARI
{T(n, k)=local(F=x, M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, x+x^2+x^3+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]} for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
Comments