

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



1, 9, 30, 20, 945, 8028, 19404, 18480, 6160, 1403325, 20545650, 94064328, 200166120, 220540320, 122522400, 27227200, 820945125, 17610977880, 124110533448, 431932849920, 857710030320, 1023307084800, 728175127680, 285558873600, 47593145600
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OFFSET

0,2


COMMENTS

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.


REFERENCES

Amparo Gil, Javier Segura and Nico M. Temme, ACM TOMS, Volume 32, Issue 1 (March 2006), pages 70101.
Amparo Gil, Javier Segura and Nico M. Temme, Numerical Methods for Special Functions, SIAM, 2007, pages 378385. See Equations 12.121 through 12.125


LINKS

Chris Kormanyos, Rows s = 0..122, flattened


FORMULA

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}] )
phi_0 = 1, phi_1 = 0


EXAMPLE

The polynomials phi_0, phi_1, phi_2 and phi_3 are:
1
(t/12) (9 + 30t + 20t^2)
(t^2/288) (945 + 8028t + 19404t^2 + 18480t^3 + 6160t^4)
(t^3/51840) (1403325 + 20545650t + 94064328t^2 + 200166120t^3 + 220540320t^4 + 122522400t^5 + 27227200t^6


MATHEMATICA

pktop = {1, 9, 30, 20};
pkbot = {1, 12};
p = (t/12) (9 + (30 t) + (20 (t^2)));
Do[pk = (4 (t^2) ((t + 1)^2)) D[p, t]  ((1/4) Integrate[((20 (t^2)) + (20 t) + 3) p, {t, 0, t}]);
p = Together[Simplify[pk]];
Do[pktop = Append[pktop, Coefficient[Expand[Numerator[p]], t^n]], {n, k, (2 k) + k, 1}];
pkbot = Append[pkbot, Denominator[p]];
Print[k], {k, 2, 10, 1}];


CROSSREFS

For denominators see A001164.
Sequence in context: A291159 A104516 A279618 * A179506 A185653 A326150
Adjacent sequences: A158500 A158501 A158502 * A158504 A158505 A158506


KEYWORD

sign,tabf


AUTHOR

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


STATUS

approved



