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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). 1

%I #12 Apr 04 2013 09:07:55

%S 1,-9,-30,-20,945,8028,19404,18480,6160,-1403325,-20545650,-94064328,

%T -200166120,-220540320,-122522400,-27227200,820945125,17610977880,

%U 124110533448,431932849920,857710030320,1023307084800,728175127680,285558873600,47593145600

%N 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 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 Amparo Gil, Javier Segura and Nico M. Temme, ACM TOMS, Volume 32, Issue 1 (March 2006), pages 70-101.

%D 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 Chris Kormanyos, <a href="/A158503/b158503.txt">Rows s = 0..122, flattened</a>

%F phi_s+1(t) = ( -4t^2(t + 1)^2 * d/dt[phi_s(t)] ) - ( (1/4) Integrate[(20T^2 + 20T + 3) phi_s(T)], {T,0,t}] )

%F phi_0 = 1, phi_-1 = 0

%e The polynomials phi_0, phi_1, phi_2 and phi_3 are:

%e 1

%e -(t/12) (9 + 30t + 20t^2)

%e (t^2/288) (945 + 8028t + 19404t^2 + 18480t^3 + 6160t^4)

%e -(t^3/51840) (1403325 + 20545650t + 94064328t^2 + 200166120t^3 + 220540320t^4 + 122522400t^5 + 27227200t^6

%t pktop = {1, -9, -30, -20};

%t pkbot = {1, 12};

%t p = (-t/12) (9 + (30 t) + (20 (t^2)));

%t 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 p = Together[Simplify[pk]];

%t Do[pktop = Append[pktop, Coefficient[Expand[Numerator[p]], t^n]], {n, k, (2 k) + k, 1}];

%t pkbot = Append[pkbot, Denominator[p]];

%t Print[k], {k, 2, 10, 1}];

%Y For denominators see A001164.

%K sign,tabf

%O 0,2

%A Chris Kormanyos (ckormanyos(AT)yahoo.com), Mar 20 2009

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