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A158503 Triangle read by rows: numerators of coefficients of the polynomials phi_s(t) used for asymptotic elementary function expansions of parabolic cylinder functions U(a, x), V(a, x).

%I A158503 #18 May 09 2025 23:11:47
%S A158503 1,-9,-30,-20,945,8028,19404,18480,6160,-1403325,-20545650,-94064328,
%T A158503 -200166120,-220540320,-122522400,-27227200,820945125,17610977880,
%U A158503 124110533448,431932849920,857710030320,1023307084800,728175127680,285558873600,47593145600
%N A158503 Triangle read by rows: numerators of coefficients of the polynomials phi_s(t) used for asymptotic elementary function expansions of parabolic cylinder functions U(a, x), V(a, x).
%C A158503 Each polynomial phi_s(t) has 2s+1 terms. The signs of the polynomials alternate with s with positive coefficients for s even and negative coefficients for s odd.
%D A158503 Amparo Gil, Javier Segura and Nico M. Temme, Computing the real parabolic cylinder functions U(a, x), V(a, x), ACM TOMS, Volume 32, Issue 1 (March 2006), pages 70-101.
%D A158503 Amparo Gil, Javier Segura and Nico M. Temme, Numerical Methods for Special Functions, SIAM, 2007, pages 378-385. See Equations 12.121 through 12.125
%H A158503 Chris Kormanyos, <a href="/A158503/b158503.txt">Rows s = 0..122, flattened</a>
%F A158503 phi_s+1(t) = ( -4*t^2*(t + 1)^2 * d/dt[phi_s(t)] ) - (1/4)*Integral_{T=0..t} (20*T^2 + 20*T + 3)*phi_s(T) dT
%F A158503 phi_0 = 1, phi_-1 = 0
%e A158503 The polynomials phi_0, phi_1, phi_2 and phi_3 are:
%e A158503   1
%e A158503   -(t/12)*(9 + 30*t + 20*t^2)
%e A158503   (t^2/288)*(945 + 8028*t + 19404*t^2 + 18480*t^3 + 6160*t^4)
%e A158503   -(t^3/51840)*(1403325 + 20545650*t + 94064328*t^2 + 200166120*t^3 + 220540320*t^4 + 122522400*t^5 + 27227200*t^6)
%t A158503 pktop = {1, -9, -30, -20};
%t A158503 pkbot = {1, 12};
%t A158503 p = (-t/12) (9 + (30 t) + (20 (t^2)));
%t A158503 Do[pk = -(4 (t^2) ((t + 1)^2)) D[p, t] - ((1/4) Integrate[((20 (t^2)) + (20 t) + 3) p, {t, 0, t}]);
%t A158503 p = Together[Simplify[pk]];
%t A158503 Do[pktop = Append[pktop, Coefficient[Expand[Numerator[p]], t^n]], {n, k, (2 k) + k, 1}];
%t A158503 pkbot = Append[pkbot, Denominator[p]];
%t A158503 Print[k], {k, 2, 10, 1}];
%Y A158503 For denominators see A001164.
%K A158503 sign,tabf
%O A158503 0,2
%A A158503 Chris Kormanyos (ckormanyos(AT)yahoo.com), Mar 20 2009