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A259462 From higher-order arithmetic progressions.

%I A259462 #15 Dec 16 2024 08:48:23
%S A259462 1,30,1200,70000,5880000,691488000,110638080000,23471078400000,
%T A259462 6454546560000000,2256222608640000000,985518035453952000000,
%U A259462 529939925428193280000000,346227417946419609600000000,271655358696421539840000000000,253338025938605687439360000000000,278215820085776765945905152000000000,356811789260008702325623357440000000000
%N A259462 From higher-order arithmetic progressions.
%C A259462 "3 over n!" in Dienger's article is A087047. A_1 is A000217. - _Georg Fischer_, Dec 16 2024
%H A259462 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 A259462 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
%F A259462 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
%p A259462 rXI := proc(n, a, d)
%p A259462         n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
%p A259462 end proc:
%p A259462 A259462 := proc(n)
%p A259462         mul(rXI(i, a, d), i=1..n+1) ;
%p A259462         coeftayl(%, d=0, 1) ;
%p A259462         coeftayl(%, a=0, n) ;
%p A259462 end proc:
%p A259462 seq(A259462(n), n=1..25) ; # _R. J. Mathar_, Jul 15 2015
%t A259462 rXI[n_, a_, d_] := n(n+1)(n+2)/6*a + (n+2)(n+1)n(n-1)/24*d;
%t A259462 A259462[n_] :=
%t A259462    Product[rXI[i, a, d], {i, 1, n + 2}] //
%t A259462    SeriesCoefficient[#, {d, 0, 1}] & //
%t A259462    SeriesCoefficient[#, {a, 0, n + 1}] & ;
%t A259462 Table[A259462[n], {n, 0, 14}] (* _Jean-François Alcover_, Apr 27 2023, after _R. J. Mathar_ *)
%Y A259462 Cf. A087047, A259463, A259464.
%K A259462 nonn
%O A259462 0,2
%A A259462 _N. J. A. Sloane_, Jun 30 2015