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

Author

N. J. A. Sloane, Jun 30 2015

Keywords

• nonn

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

• 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, A006472, A259459, A378234.

Programs

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 *)

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)

Extensions

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