A259464 -id:A259464 - cp's OEIS frontend

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

N. J. A. Sloane, Jun 30 2015

nonn

"3 over n!" in Dienger's article is A087047. A_1 is A000217. - Georg Fischer, Dec 16 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. A087047, A259463, A259464.

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

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 *)

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

A259463 From higher-order arithmetic progressions.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jun 30 2015

nonn

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

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

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 *)

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

Corrected by Jean-François Alcover, May 02 2023

cp's OEIS Frontend

A259462 From higher-order arithmetic progressions.

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