A259460 -id:A259460 - cp's OEIS frontend

A259459 From higher-order arithmetic progressions.

1, 18, 360, 9000, 283500, 11113200, 533433600, 30862944000, 2121827400000, 171160743600000, 16020645600960000, 1722947613266880000, 211061082625192800000, 29223842209642080000000, 4542220046298654720000000, 787620956028186728448000000

Offset: 0

N. J. A. Sloane, Jun 30 2015

nonn

The expression "2 over n!" in the article is A006472(n+1). It is used in A259459 - A378234 (C_1 - C_3) on page 13. - Georg Fischer, Dec 06 2024

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

Cf. A006472, A259460, A378234.

rV := proc(n,a,d)
        n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259459 := proc(n)
        mul(rV(i,a,d),i=1..n+1) ;
        coeftayl(%,d=0,1) ;
        coeftayl(%,a=0,n) ;
end proc:
seq(A259459(n),n=1..15) ; # R. J. Mathar, Jul 14 2015

rV[n_, a_, d_] := n(n+1)/2*a + (n-1)n(n+1)/6*d;
A259459[n_] :=
   Product[rV[i, a, d], {i, 1, n+2}] //
   SeriesCoefficient[#, {d, 0, 1}]& //
   SeriesCoefficient[#, {a, 0, n+1}]&;
Table[A259459[n], {n, 0, 14}] (* Jean-François Alcover, Apr 27 2023, after R. J. Mathar *)

-2*n*a(n) +(n+3)*(n+2)^2*a(n-1)=0. - R. J. Mathar, Jul 15 2015
Conjectured g.f.: 3F0(4,3,3;;x/2). - R. J. Mathar, Aug 09 2015
a(n) = (n+3)!*(n+2)!/2^(n+2)*(n+1)*(n+2)/6. - Georg Fischer, Dec 06 2024

A378234 From higher-order arithmetic progressions: Corrected version of A259461.

Original entry on oeis.org

40, 5000, 472500, 43218000, 4148928000, 432081216000, 49509306000000, 6275893932000000, 881135508052800000, 136878615942868800000, 23474682634201999200000, 4432282735129048800000000, 918537831584839065600000000, 208281986149676045967360000000, 51516317681413623440962560000000

Offset: 0

Georg Fischer, Dec 16 2024

nonn

Only the first 5 terms of A259461 are correct. - R. J. Mathar, Jul 14 2015

"2 over n!" on page 13 in the Dienger article is A006472; A_3 is A001303.

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

Cf. A001303, A006472, A259459, A259460, A259461.

rV := proc(n,a,d)
    n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
end proc:
A259461 := proc(n)
    mul(rV(i,a,d),i=1..n+3) ;
    coeftayl(%,d=0,3) ;
    coeftayl(%,a=0,n) ;
end proc:
seq(A259461(n),n=1..5) ; # R. J. Mathar, Jul 14 2015

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

a(n) = (n+5)!*(n+4)!^3 / (1296*2^(n+4)*n!^2*(n+2)*(n+1)).

cp's OEIS Frontend

A259459 From higher-order arithmetic progressions.

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