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A259457 From higher-order arithmetic progressions.

%I A259457 #13 Dec 07 2024 07:56:01
%S A259457 3,66,1050,15300,220500,3245760,49533120,789264000,13172544000,
%T A259457 230519520000,4229703878400,81315551116800,1636227552960000,
%U A259457 34417989365760000,755835784704000000,17305616126582784000,412559358036553728000,10227311816872550400000,263309943217447526400000,7032029553158658048000000
%N A259457 From higher-order arithmetic progressions.
%H A259457 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 A259457 Conjecture: 3*n*a(n) +(-3*n^2-19*n-44)*a(n-1) -2*(n+2)^2*a(n-2)=0. - _R. J. Mathar_, Jul 15 2015
%F A259457 From _Georg Fischer_, Dec 06 2024: (Start)
%F A259457 a(n) = (n+3)!*(n+2)*(n+1)*(n+3)*(3*n+8)/96.
%F A259457 D-finite with recurrence: -n*(3*n+5)*a(n) + (n+3)^2*(3*n+8)*a(n-1) = 0. (End)
%p A259457 rX := proc(n, a, d)
%p A259457         n*a+(n-1)*n/2*d;
%p A259457 end proc:
%p A259457 A259457 := proc(n)
%p A259457         mul(rX(i, a, d), i=1..n+2) ;
%p A259457         coeftayl(%, d=0, 2) ;
%p A259457         coeftayl(%, a=0, n) ;
%p A259457 end proc:
%p A259457 seq(A259457(n), n=1..25) ; # _R. J. Mathar_, Jul 15 2015
%t A259457 rX[n_, a_, d_] := n*a + (n-1)*n/2*d;
%t A259457 A259457[n_] :=
%t A259457    Product[rX[i, a, d], {i, 1, n+3}]//
%t A259457    SeriesCoefficient[#, {d, 0, 2}]&//
%t A259457    SeriesCoefficient[#, {a, 0, n+1}]&;
%t A259457 Table[A259457[n], {n, 0, 17}] (* _Jean-François Alcover_, Apr 27 2023, after _R. J. Mathar_ *)
%Y A259457 Cf. A000914, A259248.
%K A259457 nonn
%O A259457 0,1
%A A259457 _N. J. A. Sloane_, Jun 30 2015