A259458 From higher-order arithmetic progressions.

18, 750, 20250, 463050, 9878400, 205752960, 4286520000, 90561240000, 1956122784000, 43410118752000, 992644715462400, 23427803599200000, 571192163942400000, 14391113340764160000, 374682915193466880000, 10078235746321526784000, 279950992953375744000000, 8026706333564126208000000

Offset: 0

Author

N. J. A. Sloane, Jun 30 2015

Keywords

• nonn

Comments

A_2 in page 12 of the article is A001303. - Georg Fischer, Dec 06 2024

Links

• Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]

Crossrefs

Cf. A001303, A259457.

Programs

Maple

rX := proc(n, a, d)
        n*a+(n-1)*n/2*d;
end proc:
A259458 := proc(n)
        mul(rX(i, a, d), i=1..n+3) ;
        coeftayl(%, d=0, 3) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259458(n), n=1..25) ; # R. J. Mathar, Jul 15 2015

Mathematica

rX[n_, a_, d_] := n*a + (n-1)*n/2*d;
A259458[n_] :=
   Product[rX[i, a, d], {i, 1, n+4}] //
   SeriesCoefficient[#, {d, 0, 3}]& //
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259458[n], {n, 0, 16}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

Formula

D-finite with recurrence: -n*(n+2)*a(n) +(n+4)^3*a(n-1)=0. - R. J. Mathar, Jul 15 2015

Conjectured g.f.: 18*3F1(5,5,5;3;x). - R. J. Mathar, Aug 09 2015

a(n) = (n+4)!*(n+1)*(n+2)*(n+3)^2*(n+4)^2/384. - Georg Fischer, Dec 06 2024