A259460 From higher-order arithmetic progressions.

4, 220, 10500, 535500, 30870000, 2044828800, 156029328000, 13673998800000, 1369285948800000, 155756276676000000, 20005336176265440000, 2884501462544301600000, 464334381775424160000000, 83021688624014300160000000, 16408769917253890176000000000, 3569104362241728159962112000000, 850861011640079911341911040000000

Offset

0,1

Comments

The expression "2 over n!" in the article is A006472(n+1). It is used in C_1, C_2, C_3 (A259459, A259460, A378234) on page 13. A_2 is A000914. - Georg Fischer, Dec 06 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

From Georg Fischer, Dec 06 2024: (Start)

a(n) = (n+4)!*(n+3)!/2^(n+3)/9 * (n+2)*(n+1)*(n+3)*(3*n+8)/24.

D-finite with recurrence: 2*n*(3*n+5)*a(n) - (n+3)^2*(n+4)*(3*n+8)*a(n-1) = 0. (End)

Maple

rV := proc(n, a, d)
    n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259460 := proc(n)
    mul(rV(i, a, d), i=1..n+2) ;
    coeftayl(%, d=0, 2) ;
    coeftayl(%, a=0, n) ;
end proc:
seq(A259460(n), n=1..18) ; # R. J. Mathar, Jul 14 2015

Mathematica

rV[n_, a_, d_] := n (n + 1)/2*a + (n - 1) n (n + 1)/6*d;
A259460[n_] :=
   Product[rV[i, a, d], {i, 1, n + 3}] //
   SeriesCoefficient[#, {d, 0, 2}] & //
   SeriesCoefficient[#, {a, 0, n + 1}] & ;
Table[A259460[n], {n, 0, 16}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

Crossrefs

Cf. A000914, A006472, A259459, A378234.

Sequence in context: A078693 A222421 A025420 * A227822 A052209 A211610

Adjacent sequences: A259457 A259458 A259459 * A259461 A259462 A259463

Keyword

nonn

Author

N. J. A. Sloane, Jun 30 2015

Extensions

Typos in data corrected by Jean-François Alcover, Apr 27 2023

Status

approved