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%I M3081 #14 Dec 26 2021 21:40:24
%S 1,3,21,151,1257,12651,151933,2127231,34035921,612646867,12252937701,
%T 269564629863,6469551117241,168208329048891,4709833213369677,
%U 141294996401091151,4521439884834917793,153728956084387206051,5534242419037939419061,210301211923441697925687
%N a(n+1) = (2n+3)*a(n) - 2n*a(n-1) + 8n, a(0) = 1, a(1) = 3.
%D M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 36. [From _N. J. A. Sloane_, Jan 29 2009]
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H D. Doster, <a href="https://www.jstor.org/stable/2974539">Problem 10403</a>, Amer. Math. Monthly, Vol. 101 (1994), p. 792; <a href="https://www.jstor.org/stable/2974594">Solution</a>, Vol. 104 (1997), p. 368.
%F a(n) = 2*n*a(n-1) + (2*n-1)^2 = 2 * floor(e^(1/2) * n! * 2^n) - (2*n+1) = 2*A010844(n) - (2n+1). - _Michael Somos_, Mar 26 1999
%e 1 + 3*x + 21*x^2 + 151*x^3 + 1257*x^4 + 12651*x^5 + 151933*x^6 + 2127231*x^7 + ...
%p a:=proc(n) option remember; if n = 0 then RETURN(1); fi; if n = 1 then RETURN(3); fi; (2*n+1)*a(n-1)-(2*n-2)*a(n-2) + 8*(n-1); end;
%Y Cf. A010844.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_