A318918 Denominators of power series solution to differential equation diff(diff(y(t),t),t)+t*y(t) = 0, with initial conditions y(0)=1, Dy(0)=1/2.

1, 2, 1, 6, 24, 1, 180, 1008, 1, 12960, 90720, 1, 1710720, 14152320, 1, 359251200, 3396556800, 1, 109930867200, 1161622425600, 1, 46170964224000, 536669560627200, 1, 25486372251648000, 322001736376320000, 1, 17891433320656896000

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..632

Arieh Iserles, Expansions that grow on trees", Notices Amer. Math. Soc., 44 (No. 4, 2002), 430-439. See p. 433.

Formula

a(n) = 1 if n == 2 (mod 3), a(n+3) = (n+2)*(n+3)*a(n) otherwise. # Robert Israel, Jul 15 2019

Example

The solution is 1 + (1/2)*t - (1/6)*t^3 - (1/24)*t^4 + (1/180)*t^6 + (1/1008)*t^7 - (1/12960)*t^9 - (1/90720)*t^10 + (1/1710720)*t^12 + ...

Maple

with(powseries); with(gfun);
e1:= diff(y(t), t$2)+t*y(t)=0; iv := y(0)=1, D(y)(0)=1/2;
soln := powsolve( ( e1, iv) ); t1:= tpsform( soln, t, 45 );
t2:=seriestolist(t1); t3:=map(denom, t2);
# Alternative:
f:= proc(n) option remember;
  if n mod 3 = 2 then return 1 fi;
  n*(n-1)*procname(n-3)
end proc:
f(0):= 1: f(1):= 2:
map(f, [$0..50]); # Robert Israel, Jul 15 2019

Mathematica

f[n_] := f[n] = Switch[n, 0, 1, 1, 2, _, If[Mod[n, 3]==2, 1, n(n-1)f[n-3]]];
Table[f[n], {n, 0, 50}] (* Jean-François Alcover, Aug 27 2022, after Robert Israel *)

Crossrefs

Sequence in context: A239740 A371986 A268371 * A100404 A355005 A320570

Adjacent sequences: A318915 A318916 A318917 * A318919 A318920 A318921

Keyword

nonn

Author

N. J. A. Sloane, Sep 05 2018

Status

approved