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%I #13 Jan 26 2024 12:46:49
%S 1,0,1,1,0,2,0,1,4,2,2,9,8,4,2,22,24,14,8,56,70,52,24,5,146,208,176,
%T 84,30,388,624,574,320,120,14,1048,1876,1868,1184,470,112,2869,5648,
%U 6088,4236,1900,560,42,7942,17040,19804,14928,7560,2492,420,22192,51526,64232
%N Coefficients of a recursive polynomial based on Chaitin's S expressions: a(0)=1; a(1)=x; a(2)=1; a(n)=vector(a(n-1)).reverse(a(n-1)).
%C The result is an alternative way to expand s expressions as a binary rooted tree recursion.
%D G. J. Chaitin, Algorithmic Information Theory, Cambridge Press, 1987, page 169
%F a(0)=1;a(1)=x;a(2)=1;
%F a(n)=vector(a(n-1)).reverse(a(n-1));
%F t(n,m)=coefficients(a(n) in x)
%F Let b(0) = b(2) = 1, b(1) = x, and b(n) = Sum_{i=1..n} b(i-1) * b(n-i) if n>2. Then T(n, k) = coefficient of x^k in b(n) where 0 <= k <= (n+1)/2.
%F G.f. A(x,y) satisfies A(x,y) = 1 - x * (1 - x - y + 2*x*y) + x * A(x,y)^2. - _Michael Somos_, Jan 09 2012
%F G.f.: ( 1 - sqrt( (1 - 2*x)^2 - 4*x^2 * (x + y - 2*x*y) )) / (2*x). - _Michael Somos_, Jan 09 2012
%F Row sums are A025262 if offset 0.
%e {1},
%e {0, 1},
%e {1, 0},
%e {2, 0, 1},
%e {4, 2, 2},
%e {9, 8, 4, 2},
%e {22, 24, 14, 8},
%e {56, 70, 52, 24, 5},
%e {146, 208, 176, 84, 30},
%e {388, 624, 574, 320, 120, 14},
%e {1048, 1876, 1868, 1184, 470, 112},
%e {2869, 5648, 6088, 4236, 1900, 560, 42},
%e {7942, 17040, 19804, 14928, 7560, 2492, 420},
%e {22192, 51526, 64232, 52208, 29190, 10864, 2520, 132},
%e {62510, 156128, 207808, 181320, 110260, 46256, 12684, 1584},
%e {177308, 473952, 670966, 625408, 410400, 190932, 59976, 11088, 429}
%t a[0] := 1; a[1] := x; a[2] = 1;
%t a[n_] := a[n] = Table[a[i], {i, 0, n - 1}].Table[a[n - 1 - i], {i, 0, n - 1}];
%t Table[ CoefficientList[a[n], x], {n, 0, 15}];
%t Flatten[%]
%o (PARI) {T(n, k) = if( 2*k-1 > n, 0, polcoeff( polcoeff( ( 1 - sqrt( (1 - 2*x)^2 - 4*x^2 * (x + y - 2*x*y) + x^2*O(x^n))) / (2*x), n), k))} /* _Michael Somos_, Jan 09 2012 */
%Y Cf. A025227, A025262, A072851, A124027
%K nonn,tabf
%O 0,6
%A _Roger L. Bagula_, Apr 24 2010
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