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

0,4

Comments

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

Links

Table of n, a(n) for n=0..25.

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

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

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 *)

Crossrefs

Cf. A025748, A073006, A108411.

Sequence in context: A062848 A054884 A061518 * A174937 A180851 A056316

Adjacent sequences: A218537 A218538 A218539 * A218541 A218542 A218543

Keyword

nonn,easy

Author

R. J. Mathar, Nov 01 2012

Status

approved