cp's OEIS Frontend

A218540 Reduced third-order Patalan numbers.

%I A218540 #16 Aug 20 2025 08:58:43
%S A218540 1,1,1,5,10,66,154,1122,2805,21505,55913,442221,1179256,9524760,
%T A218540 25852920,211993944,582983346,4835332458,13431479050,112400272050,
%U A218540 314720761740,2652646420380,7475639911980,63380425340700,179577871798650,1530003467724498
%N A218540 Reduced third-order Patalan numbers.
%C A218540 Obtained by removing powers of 3 in a systematic manner from the Patalan numbers A025748.
%H A218540 Richard J. Mathar, <a href="https://arxiv.org/abs/1211.3963">Series expansion of generalized Fresnel integrals</a>, arXiv:1211.3963 [math.CA], 2012, eq. (4.19).
%F A218540 a(n) = A025748(n)/A108411(n).
%F A218540 D-finite with recurrence n*(n+2)*(n-1)*a(n) + (n-1)*(n-2)*(n+4)*a(n-1) - 3*(3*n-4)*(3*n-7)*(n+2)*a(n-2) - 3*(3*n-10)*(n+4)*(3*n-7)*a(n-3) = 0, n >= 4.
%F A218540 a(n) ~ 3^(2*n-2-floor(n/2)) / (Gamma(2/3) * n^(4/3)). - _Amiram Eldar_, Aug 20 2025
%p A218540 A218540 := proc(n)
%p A218540     option remember;
%p A218540     if n <=2 then
%p A218540         1;
%p A218540     elif n = 3 then
%p A218540         5 ;
%p A218540     else
%p A218540         (n-1)*(n-2)*(n+4)*procname(n-1)-3*(3*n-4)*(3*n-7)*(n+2)*procname(n-2)-3*(3*n-10)*(n+4)*(3*n-7)*procname(n-3) ;
%p A218540         -%/n/(n+2)/(n-1) ;
%p A218540     end if;
%p A218540 end proc:
%t A218540 a[n_] := 3^(2*n-2-Floor[n/2]) * Pochhammer[2/3, n-1]/n!; a[0] = 1; Array[a, 26, 0] (* _Amiram Eldar_, Aug 20 2025 *)
%Y A218540 Cf. A025748, A073006, A108411.
%K A218540 nonn,easy
%O A218540 0,4
%A A218540 _R. J. Mathar_, Nov 01 2012