A259457 From higher-order arithmetic progressions.

3, 66, 1050, 15300, 220500, 3245760, 49533120, 789264000, 13172544000, 230519520000, 4229703878400, 81315551116800, 1636227552960000, 34417989365760000, 755835784704000000, 17305616126582784000, 412559358036553728000, 10227311816872550400000, 263309943217447526400000, 7032029553158658048000000

Offset: 0

Author

N. J. A. Sloane, Jun 30 2015

Keywords

• nonn

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. A000914, A259248.

Programs

Maple

rX := proc(n, a, d)
        n*a+(n-1)*n/2*d;
end proc:
A259457 := proc(n)
        mul(rX(i, a, d), i=1..n+2) ;
        coeftayl(%, d=0, 2) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259457(n), n=1..25) ; # R. J. Mathar, Jul 15 2015

Mathematica

rX[n_, a_, d_] := n*a + (n-1)*n/2*d;
A259457[n_] :=
   Product[rX[i, a, d], {i, 1, n+3}]//
   SeriesCoefficient[#, {d, 0, 2}]&//
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259457[n], {n, 0, 17}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

Formula

Conjecture: 3*n*a(n) +(-3*n^2-19*n-44)*a(n-1) -2*(n+2)^2*a(n-2)=0. - R. J. Mathar, Jul 15 2015

From Georg Fischer, Dec 06 2024: (Start)

a(n) = (n+3)!*(n+2)*(n+1)*(n+3)*(3*n+8)/96.

D-finite with recurrence: -n*(3*n+5)*a(n) + (n+3)^2*(3*n+8)*a(n-1) = 0. (End)