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A259460 From higher-order arithmetic progressions.

%I A259460 #22 Dec 17 2024 03:19:14
%S A259460 4,220,10500,535500,30870000,2044828800,156029328000,13673998800000,
%T A259460 1369285948800000,155756276676000000,20005336176265440000,
%U A259460 2884501462544301600000,464334381775424160000000,83021688624014300160000000,16408769917253890176000000000,3569104362241728159962112000000,850861011640079911341911040000000
%N A259460 From higher-order arithmetic progressions.
%C A259460 The expression "2 over n!" in the article is A006472(n+1). It is used in C_1, C_2, C_3 (A259459, A259460, A378234) on page 13. A_2 is A000914. - _Georg Fischer_, Dec 06 2024
%H A259460 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 A259460 From _Georg Fischer_, Dec 06 2024: (Start)
%F A259460 a(n) = (n+4)!*(n+3)!/2^(n+3)/9 * (n+2)*(n+1)*(n+3)*(3*n+8)/24.
%F A259460 D-finite with recurrence: 2*n*(3*n+5)*a(n) - (n+3)^2*(n+4)*(3*n+8)*a(n-1) = 0. (End)
%p A259460 rV := proc(n,a,d)
%p A259460     n*(n+1)/2*a+(n-1)*n*(n+1)/6*d;
%p A259460 end proc:
%p A259460 A259460 := proc(n)
%p A259460     mul(rV(i,a,d),i=1..n+2) ;
%p A259460     coeftayl(%,d=0,2) ;
%p A259460     coeftayl(%,a=0,n) ;
%p A259460 end proc:
%p A259460 seq(A259460(n),n=1..18) ; # _R. J. Mathar_, Jul 14 2015
%t A259460 rV[n_, a_, d_] := n (n + 1)/2*a + (n - 1) n (n + 1)/6*d;
%t A259460 A259460[n_] :=
%t A259460    Product[rV[i, a, d], {i, 1, n + 3}] //
%t A259460    SeriesCoefficient[#, {d, 0, 2}] & //
%t A259460    SeriesCoefficient[#, {a, 0, n + 1}] & ;
%t A259460 Table[A259460[n], {n, 0, 16}] (* _Jean-François Alcover_, Apr 27 2023, after _R. J. Mathar_ *)
%Y A259460 Cf. A000914, A006472, A259459, A378234.
%K A259460 nonn
%O A259460 0,1
%A A259460 _N. J. A. Sloane_, Jun 30 2015
%E A259460 Typos in data corrected by _Jean-François Alcover_, Apr 27 2023