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A259459 From higher-order arithmetic progressions.

%I A259459 #26 Dec 17 2024 03:18:49
%S A259459 1,18,360,9000,283500,11113200,533433600,30862944000,2121827400000,
%T A259459 171160743600000,16020645600960000,1722947613266880000,
%U A259459 211061082625192800000,29223842209642080000000,4542220046298654720000000,787620956028186728448000000
%N A259459 From higher-order arithmetic progressions.
%C A259459 The expression "2 over n!" in the article is A006472(n+1). It is used in A259459 - A378234 (C_1 - C_3) on page 13. - _Georg Fischer_, Dec 06 2024
%H A259459 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 A259459 -2*n*a(n) +(n+3)*(n+2)^2*a(n-1)=0. - _R. J. Mathar_, Jul 15 2015
%F A259459 Conjectured g.f.: 3F0(4,3,3;;x/2). - _R. J. Mathar_, Aug 09 2015
%F A259459 a(n) = (n+3)!*(n+2)!/2^(n+2)*(n+1)*(n+2)/6. - _Georg Fischer_, Dec 06 2024
%p A259459 rV := proc(n,a,d)
%p A259459         n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
%p A259459 end proc:
%p A259459 A259459 := proc(n)
%p A259459         mul(rV(i,a,d),i=1..n+1) ;
%p A259459         coeftayl(%,d=0,1) ;
%p A259459         coeftayl(%,a=0,n) ;
%p A259459 end proc:
%p A259459 seq(A259459(n),n=1..15) ; # _R. J. Mathar_, Jul 14 2015
%t A259459 rV[n_, a_, d_] := n(n+1)/2*a + (n-1)n(n+1)/6*d;
%t A259459 A259459[n_] :=
%t A259459    Product[rV[i, a, d], {i, 1, n+2}] //
%t A259459    SeriesCoefficient[#, {d, 0, 1}]& //
%t A259459    SeriesCoefficient[#, {a, 0, n+1}]&;
%t A259459 Table[A259459[n], {n, 0, 14}] (* _Jean-François Alcover_, Apr 27 2023, after _R. J. Mathar_ *)
%Y A259459 Cf. A006472, A259460, A378234.
%K A259459 nonn
%O A259459 0,2
%A A259459 _N. J. A. Sloane_, Jun 30 2015