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A378234 From higher-order arithmetic progressions: Corrected version of A259461.

%I A378234 #21 Dec 17 2024 03:24:11
%S A378234 40,5000,472500,43218000,4148928000,432081216000,49509306000000,
%T A378234 6275893932000000,881135508052800000,136878615942868800000,
%U A378234 23474682634201999200000,4432282735129048800000000,918537831584839065600000000,208281986149676045967360000000,51516317681413623440962560000000
%N A378234 From higher-order arithmetic progressions: Corrected version of A259461.
%C A378234 Only the first 5 terms of A259461 are correct. - _R. J. Mathar_, Jul 14 2015
%C A378234 "2 over n!" on page 13 in the Dienger article is A006472; A_3 is A001303.
%H A378234 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 A378234 D-finite with recurrence: -2*n*(n+2)*a(n) + (n+4)^3*(n+5)*a(n-1) = 0.
%F A378234 a(n) = (n+5)!*(n+4)!^3 / (1296*2^(n+4)*n!^2*(n+2)*(n+1)).
%p A378234 rV := proc(n,a,d)
%p A378234     n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
%p A378234 end proc:
%p A378234 A259461 := proc(n)
%p A378234     mul(rV(i,a,d),i=1..n+3) ;
%p A378234     coeftayl(%,d=0,3) ;
%p A378234     coeftayl(%,a=0,n) ;
%p A378234 end proc:
%p A378234 seq(A259461(n),n=1..5) ; # _R. J. Mathar_, Jul 14 2015
%Y A378234 Cf. A001303, A006472, A259459, A259460, A259461.
%K A378234 nonn
%O A378234 0,1
%A A378234 _Georg Fischer_, Dec 16 2024