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A166900 Triangle, read by rows, that transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).

%I A166900 #6 Jan 19 2014 18:54:54
%S A166900 1,1,1,2,4,1,9,21,9,1,64,156,84,16,1,630,1540,935,230,25,1,7916,19160,
%T A166900 12480,3564,510,36,1,121023,288813,196623,61845,10465,987,49,1,
%U A166900 2179556,5123608,3591560,1207696,228800,25864,1736,64,1,45179508,104657520
%N A166900 Triangle, read by rows, that transforms rows into diagonals in the table of coefficients of successive iterations of x+x^2 (cf. A122888).
%C A166900  Compare to the triangle A071207 that transforms rows into diagonals in the table of iterations of x/(1-x), where A071207(n,k) gives the number of labeled free trees with n vertices and k children of the root that have a label smaller than the label of the root. Does this triangle have a similar interpretation?
%H A166900 Paul D. Hanna, <a href="/A166900/b166900.txt">Table of n, a(n) for n=0..495 (rows 0..30)</a>
%e A166900 Triangle begins:
%e A166900 1;
%e A166900 1, 1;
%e A166900 2, 4, 1;
%e A166900 9, 21, 9, 1;
%e A166900 64, 156, 84, 16, 1;
%e A166900 630, 1540, 935, 230, 25, 1;
%e A166900 7916, 19160, 12480, 3564, 510, 36, 1;
%e A166900 121023, 288813, 196623, 61845, 10465, 987, 49, 1;
%e A166900 2179556, 5123608, 3591560, 1207696, 228800, 25864, 1736, 64, 1;
%e A166900 45179508, 104657520, 74847168, 26415840, 5426949, 695079, 56511, 2844, 81, 1;
%e A166900 1059312264, 2420186616, 1755406674, 642448632, 140247810, 19683060, 1830080, 112520, 4410, 100, 1; ...
%e A166900 Coefficients in self-compositions of (x+x^2) form table A122888:
%e A166900 1;
%e A166900 1, 1;
%e A166900 1, 2, 2, 1;
%e A166900 1, 3, 6, 9, 10, 8, 4, 1;
%e A166900 1, 4, 12, 30, 64, 118, 188, 258, 302, 298, 244, 162, 84, 32, 8, 1;
%e A166900 1, 5, 20, 70, 220, 630, 1656, 4014, 8994, 18654, 35832, 63750,...;
%e A166900 1, 6, 30, 135, 560, 2170, 7916, 27326, 89582, 279622, 832680,...; ...
%e A166900 This triangle T transforms rows of A122888 into diagonals of A122888;
%e A166900 the initial diagonals begin:
%e A166900 A112319: [1, 1, 2, 9, 64, 630, 7916, 121023, 2179556, 45179508, ...];
%e A166900 A112317: [1, 2, 6, 30, 220, 2170, 27076, 409836, 7303164, 149837028,..];
%e A166900 A112320: [1, 3, 12, 70, 560, 5810, 74760, 1153740, 20817588, 430604724, ...].
%e A166900 For example:
%e A166900 T * [1, 0, 0, 0, 0, 0, 0,...]~ = A112319;
%e A166900 T * [1, 1, 0, 0, 0, 0, 0,...]~ = A112317;
%e A166900 T * [1, 2, 2, 1, 0, 0, 0,...]~ = A112320.
%o A166900 (PARI) {T(n, k)=local(F=x, M, N, P, m=n); M=matrix(m+2, m+2, r, c, F=x;for(i=1, r+c-2, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, F=x;for(i=1, r, F=subst(F, x, x+x^2+x*O(x^(m+2)))); polcoeff(F, c)); P=matrix(m+1, m+1, r, c, M[r+1, c]); (P~*N~^-1)[n+1, k+1]}
%o A166900 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%Y A166900 Cf. A112319, A166901, A166902, A166903, A122888, A135080.
%K A166900 nonn,tabl
%O A166900 0,4
%A A166900 _Paul D. Hanna_, Nov 27 2009