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A167576 The first column of the ED3 array A167572.

%I A167576 #40 Feb 10 2022 07:51:38
%S A167576 1,5,23,167,1473,16413,211479,3192975,54010305,1030249845,21566327895,
%T A167576 497334999735,12405876372225,335591130336525,9716331072597975,
%U A167576 301633179343890975,9941514351641143425,348336799875365041125
%N A167576 The first column of the ED3 array A167572.
%C A167576 Basically a(n) measures the difference between the Euler factorial n! and the Luschny factorial L(n) at half-integer values. For the Luschny factorial see the link. The formula given in the Maple section is a variant of a formula given by _Cyril Damamme_ in A135457. - _Peter Luschny_, Jul 18 2015
%H A167576 G. C. Greubel, <a href="/A167576/b167576.txt">Table of n, a(n) for n = 1..400</a>
%H A167576 Peter Luschny, <a href="http://www.luschny.de/math/factorial/hadamard/HadamardsGammaFunctionMJ.html">Is the Gamma function misdefined? Hadamard versus Euler - Who found the better Gamma function?</a>
%F A167576 a(n) = (-1)^n*(2*n-3)!!*(1 + (4*n-2)*Sum_{k=0..n-1} (-1)^(k+n)/(2*k+1)).
%F A167576 a(n) = (2*n-1)*a(n-1) + 2*(-1)^n*(2*n-5)!! with a(1) = 1.
%F A167576 a(n) = 4*a(n-1) + (4*n^2 - 16*n + 15)*a(n-2) with a(1) = 1 and a(2) = 5 [Superseeker].
%F A167576 0 = a(n)*a(n+1)*(-440*a(n+2) - 220*a(n+3) + 55*a(n+4)) + a(n)*a(n+2)*(536*a(n+2) - 118*a(n+3) - 4*a(n+4)) + a(n)*a(n+3)*(-4*a(n+3) + a(n+4)) + a(n+1)^2*(-220*a(n+2) - 32*a(n+3) + 8*a(n+4)) + a(n+1)*a(n+2)*(+71*a(n+2) + 4*a(n+3) - 2*a(n+4)) + a(n+2)^2*(-4*a(n+2) + a(n+3)) if n>0. - _Michael Somos_, Jul 19 2015
%F A167576 a(n) = (-1 + (n-1/2)*LerchPhi(-1,1,n+1/2) + (-n+1/2)*LerchPhi(-1,1,-n+1/2))/(1-2*n)!!. - _Johannes W. Meijer_, Jul 20 2015
%F A167576 a(n) = A024199(n) + A135457(n). - _Cyril Damamme_, Jul 22 2015
%F A167576 a(n) = ((-1)^n/(2*n - 1) + Pi/2 - (-1)^n LerchPhi(-1, 1, n + 1/2)) (2*n - 1)!!. - _Michael Somos_, Jan 31 2019
%e A167576 G.f. = x + 5*x^2 + 23*x^3 + 167*x^4 + 1473*x^5 + 16413*x^6 + ...
%p A167576 L := x -> (1+x*(Psi(1-x/2)-Psi(1/2-x/2)))/(-x)!:
%p A167576 a := x -> (L(x-1/2)-(x-1/2)!)*2^(x-1)*sqrt(Pi):
%p A167576 seq(simplify(a(n)),n=1..18); # _Peter Luschny_, Jul 18 2015
%p A167576 a := proc(n) option remember: if n=1 then 1 else (2*n-1)*a(n-1)+2*(-1)^n*doublefactorial(2*n-5) fi: end: seq(a(n),n=1..18); # _Johannes W. Meijer_, Jul 20 2015
%t A167576 a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n - I (4 n - 2) Sum[ I^k / k, {k, 1, 2 n - 1, 2}])]; (* _Michael Somos_, Jul 20 2015 *)
%t A167576 a[ n_] := If[ n < 1, 0, (2 n - 3)!! ((-1)^n + (4 n - 2) Sum[ KroneckerSymbol[ -4, k]/ k, {k, 2 n - 1}])]; (* _Michael Somos_, Jan 31 2019 *)
%o A167576 (PARI) {a(n) = if( n<1, 0, prod(k=1, n-1, 2*k - 1) * ((-1)^n - (4*n - 2) * sum(k=1, n, (-1)^k / (2*k - 1))))}; /* _Michael Somos_, Jul 20 2015 */
%Y A167576 Equals the first column of the ED3 array A167572.
%Y A167576 Equals the first right hand column of A167583.
%Y A167576 Other columns are A167577 and A167578.
%Y A167576 Cf. A097801 (the 2*(-1)^n*(2*n-5)!! factor).
%Y A167576 Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor).
%Y A167576 Cf. A024199 and A135457.
%K A167576 easy,nonn
%O A167576 1,2
%A A167576 _Johannes W. Meijer_, Nov 10 2009