cp's OEIS Frontend

A259463 From higher-order arithmetic progressions.

%I A259463 #17 Dec 16 2024 14:40:19
%S A259463 5,550,61250,8330000,1440600000,318084480000,88994505600000,
%T A259463 31196975040000000,13537335651840000000,7186069008518400000000,
%U A259463 4614893517270516480000000,3548831033950800998400000000,3237226357799023349760000000000,3472842105575052965314560000000000
%N A259463 From higher-order arithmetic progressions.
%C A259463 "3 over n!" on page 15 in the Dienger article is A087047; A_2 is A000914. - _Georg Fischer_, Dec 16 2024
%H A259463 Karl Dienger, <a href="/A000217/a000217.pdf">Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung</a>, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
%F A259463 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
%F A259463 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
%p A259463 rXI := proc(n, a, d)
%p A259463         n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
%p A259463 end proc:
%p A259463 A259463 := proc(n)
%p A259463         mul(rXI(i, a, d), i=1..n+2) ;
%p A259463         coeftayl(%, d=0, 2) ;
%p A259463         coeftayl(%, a=0, n) ;
%p A259463 end proc:
%p A259463 seq(A259463(n), n=1..25) ; # _R. J. Mathar_, Jul 15 2015
%t A259463 rXI[n_, a_, d_] := n(n+1)(n+2)/6 a + (n+2)(n+1)n(n-1)/24 d;
%t A259463 A259463[n_] := Product[rXI[i, a, d], {i, 1, n+3}]//
%t A259463    SeriesCoefficient[#, {d, 0, 2}]&//
%t A259463    SeriesCoefficient[#, {a, 0, n+1}]&;
%t A259463 Table[A259463[n], {n, 0, 13}] (* _Jean-François Alcover_, May 02 2023, after _R. J. Mathar_ *)
%Y A259463 Cf. A000914, A087047, A259462, A259464.
%K A259463 nonn
%O A259463 0,1
%A A259463 _N. J. A. Sloane_, Jun 30 2015
%E A259463 Corrected by _Jean-François Alcover_, May 02 2023