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A136667 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Hermite polynomial of the Hochstadt kind (A137286) as related to the generalized Chebyshev in a Shabat way (A123583): p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p(x,n)^2.

%I A136667 #16 Feb 26 2018 09:32:58
%S A136667 0,1,0,-1,-3,0,4,0,-1,1,0,-25,0,10,0,-1,-63,0,144,0,-97,0,18,0,-1,1,0,
%T A136667 -1089,0,924,0,-262,0,28,0,-1,-2303,0,8352,0,-9489,0,3576,0,-574,0,40,
%U A136667 0,-1,1,0,-77841,0,103230,0,-49291,0,10548,0,-1099,0,54,0,-1,-147455,0,748800,0,-1215585,0,699630,0,-188043,0
%N A136667 Triangle read by rows: T(n, k) is the coefficient of x^k in the polynomial 1 - T_n(x)^2, where T_n(x) is the n-th Hermite polynomial of the Hochstadt kind (A137286) as related to the generalized Chebyshev in a Shabat way (A123583): p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p(x,n)^2.
%C A136667 Row sums are {0, 0, 0, -15, 1, -399, -399, -14399, -78399, -639999, -12959999}.
%D A136667 Defined: page 8 and pages 42 - 43: Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
%D A136667 G. B. Shabat and I. A. Voevodskii, Drawing curves over number fields, The Grothendieck Festschift, vol. 3, Birkhäuser, 1990, pp. 199-22
%H A136667 G. B. Shabat and A. Zvonkin, <a href="https://www.labri.fr/perso/zvonkin/Research/shabzvon.pdf">Plane trees and algebraic numbers</a>, Contemporary Math., 1994, vol. 178, 233-275.
%F A136667 out=1-A137286(x,n)^2; p(x,n)=x*p(x,n-1)-p(x,n-2); q(x,n)=1-p(x,n)^2.
%e A136667 The irregular triangle begins
%e A136667   {0},
%e A136667   {1, 0, -1},
%e A136667   {-3, 0, 4, 0, -1},
%e A136667   {1, 0, -25, 0, 10, 0, -1},
%e A136667   {-63, 0, 144, 0, -97, 0, 18, 0, -1},
%e A136667   {1, 0, -1089, 0, 924, 0, -262,0, 28, 0, -1},
%e A136667   {-2303, 0, 8352, 0, -9489, 0, 3576, 0, -574, 0, 40, 0, -1},
%e A136667   {1, 0, -77841, 0, 103230, 0, -49291, 0, 10548,0, -1099, 0, 54, 0, -1},
%e A136667   ...
%t A136667 P[x, 0] = 1; P[x, 1] = x; P[x_, n_] := P[x, n] = x*P[x, n - 1] - n*P[x, n - 2]; Q[x_, n_] := Q[x, n] = 1 - P[x, n]^2; Table[ExpandAll[Q[x, n]], {n, 0, 10}]; a = Join[{{0}}, Table[CoefficientList[Q[x, n], x], {n, 0, 10}]]; Flatten[a]
%o A136667 (PARI) polx(n) = if (n == 0, 1, if (n == 1, x, x*polx(n - 1) - n*polx(n - 2)));
%o A136667 tabf(nn) = {for (n = 0, nn, pol = 1 - polx(n)^2; for (i = 0, 2*n, print1(polcoeff(pol, i), ", "); ); print(); ); }  \\ _Michel Marcus_, Feb 26 2018
%Y A136667 Cf. A123583, A137286, A136247.
%K A136667 uned,tabf,sign
%O A136667 1,5
%A A136667 _Roger L. Bagula_, Apr 02 2008
%E A136667 Keyword changed to tabf by _Michel Marcus_, Feb 26 2018