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A259458 From higher-order arithmetic progressions.

%I A259458 #16 Dec 07 2024 07:58:52
%S A259458 18,750,20250,463050,9878400,205752960,4286520000,90561240000,
%T A259458 1956122784000,43410118752000,992644715462400,23427803599200000,
%U A259458 571192163942400000,14391113340764160000,374682915193466880000,10078235746321526784000,279950992953375744000000,8026706333564126208000000
%N A259458 From higher-order arithmetic progressions.
%C A259458 A_2 in page 12 of the article is A001303. - _Georg Fischer_, Dec 06 2024
%H A259458 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 A259458 D-finite with recurrence: -n*(n+2)*a(n) +(n+4)^3*a(n-1)=0. - _R. J. Mathar_, Jul 15 2015
%F A259458 Conjectured g.f.: 18*3F1(5,5,5;3;x). - _R. J. Mathar_, Aug 09 2015
%F A259458 a(n) = (n+4)!*(n+1)*(n+2)*(n+3)^2*(n+4)^2/384. - _Georg Fischer_, Dec 06 2024
%p A259458 rX := proc(n, a, d)
%p A259458         n*a+(n-1)*n/2*d;
%p A259458 end proc:
%p A259458 A259458 := proc(n)
%p A259458         mul(rX(i, a, d), i=1..n+3) ;
%p A259458         coeftayl(%, d=0, 3) ;
%p A259458         coeftayl(%, a=0, n) ;
%p A259458 end proc:
%p A259458 seq(A259458(n), n=1..25) ; # _R. J. Mathar_, Jul 15 2015
%t A259458 rX[n_, a_, d_] := n*a + (n-1)*n/2*d;
%t A259458 A259458[n_] :=
%t A259458    Product[rX[i, a, d], {i, 1, n+4}] //
%t A259458    SeriesCoefficient[#, {d, 0, 3}]& //
%t A259458    SeriesCoefficient[#, {a, 0, n+1}]&;
%t A259458 Table[A259458[n], {n, 0, 16}] (* _Jean-François Alcover_, Apr 27 2023, after _R. J. Mathar_ *)
%Y A259458 Cf. A001303, A259457.
%K A259458 nonn
%O A259458 0,1
%A A259458 _N. J. A. Sloane_, Jun 30 2015