A259462 From higher-order arithmetic progressions.

1, 30, 1200, 70000, 5880000, 691488000, 110638080000, 23471078400000, 6454546560000000, 2256222608640000000, 985518035453952000000, 529939925428193280000000, 346227417946419609600000000, 271655358696421539840000000000, 253338025938605687439360000000000, 278215820085776765945905152000000000, 356811789260008702325623357440000000000

Offset

0,2

Comments

"3 over n!" in Dienger's article is A087047. A_1 is A000217. - Georg Fischer, Dec 16 2024

Links

Table of n, a(n) for n=0..16.

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

Formula

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

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

Maple

rXI := proc(n, a, d)
        n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
end proc:
A259462 := proc(n)
        mul(rXI(i, a, d), i=1..n+1) ;
        coeftayl(%, d=0, 1) ;
        coeftayl(%, a=0, n) ;
end proc:
seq(A259462(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;
A259462[n_] :=
   Product[rXI[i, a, d], {i, 1, n + 2}] //
   SeriesCoefficient[#, {d, 0, 1}] & //
   SeriesCoefficient[#, {a, 0, n + 1}] & ;
Table[A259462[n], {n, 0, 14}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

Crossrefs

Cf. A087047, A259463, A259464.

Sequence in context: A075187 A265101 A214820 * A269682 A269471 A060076

Adjacent sequences: A259459 A259460 A259461 * A259463 A259464 A259465

Keyword

nonn,changed

Author

N. J. A. Sloane, Jun 30 2015

Status

approved