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A166884 Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x+x^2+x^3 (cf. A166880).

%I A166884 #9 Sep 13 2013 18:53:50
%S A166884 1,1,1,3,2,1,15,9,3,1,114,62,18,4,1,1159,593,157,30,5,1,14838,7266,
%T A166884 1812,316,45,6,1,229401,108720,25989,4271,555,63,7,1,4159662,1922166,
%U A166884 445255,70180,8595,890,84,8,1,86580636,39212154,8865333,1354750,159171,15534,1337,108,9,1
%N A166884 Triangle, read by rows, that transforms diagonals in the table of coefficients of successive iterations of x+x^2+x^3 (cf. A166880).
%H A166884 Paul D. Hanna, <a href="/A166884/b166884.txt">a(n) for n = 0..350</a> (rows 0..25, flattened).
%e A166884 This triangle begins:
%e A166884 1;
%e A166884 1, 1;
%e A166884 3, 2, 1;
%e A166884 15, 9, 3, 1;
%e A166884 114, 62, 18, 4, 1;
%e A166884 1159, 593, 157, 30, 5, 1;
%e A166884 14838, 7266, 1812, 316, 45, 6, 1;
%e A166884 229401, 108720, 25989, 4271, 555, 63, 7, 1;
%e A166884 4159662, 1922166, 445255, 70180, 8595, 890, 84, 8, 1;
%e A166884 86580636, 39212154, 8865333, 1354750, 159171, 15534, 1337, 108, 9, 1;
%e A166884 2034850425, 906623004, 201058614, 30000676, 3418245, 320070, 25963, 1912, 135, 10, 1;
%e A166884 53303009286, 23429034168, 5114874693, 748896765, 83336385, 7568355, 589057, 40882, 2631, 165, 11, 1; ...
%e A166884 Triangle A166880 of coefficients in iterations of x+x^2+x^3 begins:
%e A166884 1;
%e A166884 1,1,1;
%e A166884 1,2,4,6,8,8,6,3,1;
%e A166884 1,3,9,24,60,138,294,579,1053,1767,2739,3924,5196,6352,7152,7389,...;
%e A166884 1,4,16,60,216,744,2460,7818,23910,70446,200160,549006,1455132,...;
%e A166884 1,5,25,120,560,2540,11220,48330,203230,835080,3355950,13200648,...;
%e A166884 1,6,36,210,1200,6720,36930,199365,1058175,5526330,28417200,...;
%e A166884 1,7,49,336,2268,15078,98826,639093,4080531,25738755,160474545,...;
%e A166884 1,8,64,504,3920,30128,228984,1722084,12821788,94556532,...; ...
%e A166884 in which this triangle transforms diagonals in A166880 into each other.
%e A166884 The initial diagonals in triangle A166880 begin:
%e A166884 A166881: [1,1,4,24,216,2540,36930,639093,12821788,292495896,...];
%e A166884 A166882: [1,2,9,60,560,6720,98826,1722084,34700940,793894860,...];
%e A166884 A166883: [1,3,16,120,1200,15078,228984,4085028,83795085,1943920935,...]; ...
%e A166884 so that, if we treat the diagonals as column vectors, we have:
%e A166884 A166884 * A166881 = A166882,
%e A166884 A166884 * A166882 = A166883.
%o A166884 (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]}
%o A166884 for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y A166884 Cf. A166880, columns: A166885, A166886, A166887; A229112 (row sums).
%K A166884 nonn,tabl
%O A166884 0,4
%A A166884 _Paul D. Hanna_, Nov 21 2009