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A076051 Sum of product of odd numbers <= n and the product of even numbers <= n.

%I A076051 #11 Jun 26 2025 10:58:38
%S A076051 2,3,5,11,23,63,153,489,1329,4785,14235,56475,181215,780255,2672145,
%T A076051 12348945,44781345,220253985,840523635,4370620275,17465201775,
%U A076051 95498916975,397983749625,2278224696825,9867844134225,58917607974225
%N A076051 Sum of product of odd numbers <= n and the product of even numbers <= n.
%H A076051 G. C. Greubel, <a href="/A076051/b076051.txt">Table of n, a(n) for n = 1..800</a>
%F A076051 a(n) = o(n)+ e(n) where; o(n)=the product of odd numbers from 1 to n e(n)=the product of even numbers from 2 to n.
%F A076051 From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002: (Start)
%F A076051 a(n) = A060696(n+1).
%F A076051 a(n) = A037223(n) + abs(A055634(n)).
%F A076051 a(n) = A037223(n) + n! / A037223(n), where A037223(n) = 2^floor(n/2) * floor(n/2)!, for n>=2.
%F A076051 a(1)=2, a(2)=3, a(3)=5, a(n) = (n-1)*a(n-2) + (n-2)!! for n >= 4.
%F A076051 E.g.f.: 1 + x + (1+x+x^2)*(exp(x^2/2)*(1+sqrt(Pi/2)*erf(x/sqrt(2)))), where erf denotes the error function. (End)
%t A076051 A037223[n_] := 2^(Floor[n/2])*(Floor[n/2])!; Table[A037223[n] + n!/A037223[n] , {n,1,50}] (* _G. C. Greubel_, May 23 2017 *)
%t A076051 With[{nn = 25}, CoefficientList[Series[1 + x + (1 + x + x^2) *(Exp[x^2/2] *(1 + Sqrt[Pi/2]*Erf[x/Sqrt[2]])), {x, 0, nn}], x] Range[0, nn]!] (* _G. C. Greubel_, May 25 2017 *)
%o A076051 (PARI) for(n=1, 50, print1(2^(floor(n/2))*(floor(n/2))! + n!/(2^(floor(n/2))*(floor(n/2))!), ", ")) \\ _G. C. Greubel_, May 23 2017
%Y A076051 Cf. A037223, A055634, A060696.
%K A076051 easy,nonn
%O A076051 1,1
%A A076051 Emrehan Halici (emrehan(AT)halici.com.tr), Oct 30 2002
%E A076051 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002
%E A076051 a(1) corrected by _G. C. Greubel_, May 23 2017