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

2, 3, 5, 11, 23, 63, 153, 489, 1329, 4785, 14235, 56475, 181215, 780255, 2672145, 12348945, 44781345, 220253985, 840523635, 4370620275, 17465201775, 95498916975, 397983749625, 2278224696825, 9867844134225, 58917607974225

Offset: 1

Author

Emrehan Halici (emrehan(AT)halici.com.tr), Oct 30 2002

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..800

Crossrefs

Cf. A037223, A055634, A060696.

Programs

Mathematica

A037223[n_] := 2^(Floor[n/2])*(Floor[n/2])!; Table[A037223[n] + n!/A037223[n] , {n,1,50}] (* G. C. Greubel, May 23 2017 *)
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 *)

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

Formula

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.

From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002: (Start)

a(n) = A060696(n+1).

a(n) = A037223(n) + abs(A055634(n)).

a(n) = A037223(n) + n! / A037223(n), where A037223(n) = 2^floor(n/2) * floor(n/2)!, for n>=2.

a(1)=2, a(2)=3, a(3)=5, a(n) = (n-1)*a(n-2) + (n-2)!! for n >= 4.

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)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

a(1) corrected by G. C. Greubel, May 23 2017