A218540 Reduced third-order Patalan numbers.

1, 1, 1, 5, 10, 66, 154, 1122, 2805, 21505, 55913, 442221, 1179256, 9524760, 25852920, 211993944, 582983346, 4835332458, 13431479050, 112400272050, 314720761740, 2652646420380, 7475639911980, 63380425340700, 179577871798650, 1530003467724498

Offset: 0

Author

R. J. Mathar, Nov 01 2012

Keywords

• nonn
• easy

Comments

Obtained by removing powers of 3 in a systematic manner from the Patalan numbers A025748.

Links

• Richard J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (4.19).

Crossrefs

Cf. A025748, A073006, A108411.

Programs

Maple

A218540 := proc(n)
    option remember;
    if n <=2 then
        1;
    elif n = 3 then
        5 ;
    else
        (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) ;
        -%/n/(n+2)/(n-1) ;
    end if;
end proc:

Mathematica

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 *)

Formula

a(n) = A025748(n)/A108411(n).

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.

a(n) ~ 3^(2*n-2-floor(n/2)) / (Gamma(2/3) * n^(4/3)). - Amiram Eldar, Aug 20 2025