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A259464 From higher-order arithmetic progressions.

%I A259464 #24 May 31 2025 05:20:30
%S A259464 75,21875,5512500,1512630000,484041600000,184834742400000,
%T A259464 84715923600000000,46534591303200000000,30489464221856640000000,
%U A259464 23681690417572387200000000,21660852835272876825600000000,23175597788788462617600000000000,28817200450516396946227200000000000
%N A259464 From higher-order arithmetic progressions.
%C A259464 "3 over n!" on page 15 in the Dienger article is A087047; A_3 is A001303. - _Georg Fischer_, Dec 16 2024
%H A259464 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 A259464 D-finite with recurrence: -6*n*(n+2)*a(n) +(n+6)*(n+5)*(n+4)^3*a(n-1)=0. - _R. J. Mathar_, Jul 15 2015
%F A259464 a(n) = 2^(-n-5)*3^(-n-4)*(n+4)!*(n+5)!*(n+6)!*(n+4)^2*(n+3)^2*(n+2)*(n+1)/3072. - _Georg Fischer_, Dec 16 2024
%p A259464 rXI := proc(n, a, d)
%p A259464         n*(n+1)*(n+2)/6*a+(n+2)*(n+1)*n*(n-1)/24*d;
%p A259464 end proc:
%p A259464 A259464 := proc(n)
%p A259464         mul(rXI(i, a, d), i=1..n+3) ;
%p A259464         coeftayl(%, d=0, 3) ;
%p A259464         coeftayl(%, a=0, n) ;
%p A259464 end proc:
%p A259464 seq(A259464(n), n=1..25) ; # _R. J. Mathar_, Jul 15 2015
%t A259464 rXI[n_, a_, d_] := (n(n+1)(n+2)/6)*a+((n+2)(n+1)n(n-1)/24)*d;
%t A259464 A259464[n_] :=
%t A259464    Product[rXI[i, a, d], {i, 1, n+4}]//
%t A259464    SeriesCoefficient[#, {d, 0, 3}]&//
%t A259464    SeriesCoefficient[#, {a, 0, n+1}]&;
%t A259464 Table[A259464[n], {n, 0, 12}] (* _Jean-François Alcover_, Apr 26 2023, after _R. J. Mathar_ *)
%Y A259464 Cf. A001303, A087047, A259462, A259463.
%K A259464 nonn
%O A259464 0,1
%A A259464 _N. J. A. Sloane_, Jun 30 2015