A259463 From higher-order arithmetic progressions.

5, 550, 61250, 8330000, 1440600000, 318084480000, 88994505600000, 31196975040000000, 13537335651840000000, 7186069008518400000000, 4614893517270516480000000, 3548831033950800998400000000, 3237226357799023349760000000000, 3472842105575052965314560000000000

Offset: 0

Author

N. J. A. Sloane, Jun 30 2015

Keywords

• nonn

Comments

"3 over n!" on page 15 in the Dienger article is A087047; A_2 is A000914. - Georg Fischer, Dec 16 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. A000914, A087047, A259462, A259464.

Programs

Maple

rXI := proc(n, a, d)
        n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
end proc:
A259463 := proc(n)
        mul(rXI(i, a, d), i=1..n+2) ;
        coeftayl(%, d=0, 2) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259463(n), n=1..25) ; # R. J. Mathar, Jul 15 2015

Mathematica

rXI[n_, a_, d_] := n(n+1)(n+2)/6 a + (n+2)(n+1)n(n-1)/24 d;
A259463[n_] := Product[rXI[i, a, d], {i, 1, n+3}]//
   SeriesCoefficient[#, {d, 0, 2}]&//
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259463[n], {n, 0, 13}] (* Jean-François Alcover, May 02 2023, after R. J. Mathar *)

Formula

D-finite with recurrence: -6*n*(3*n+5)*a(n) +(n+5)*(n+4)*(3*n+8)*(n+3)^2*a(n-1)=0. - R. J. Mathar, Jul 15 2015

a(n) = 2^(-n-4)*3^(-n-3)*(n+3)!*(n+4)!*(n+5)!*(n+2)*(n+1)*(n+3)*(3*n+8)/384. - Georg Fischer, Dec 16 2024

Extensions

Corrected by Jean-François Alcover, May 02 2023