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A114160 E.g.f. is A(x) = (1-log(B(x)))/B(x), where B(x) = sqrt(1-2*x).

%I A114160 #26 Jan 21 2025 06:31:06
%S A114160 1,2,7,38,281,2634,29919,399342,6125265,106156530,2051433495,
%T A114160 43734832470,1019650457385,25807495577850,704708234182575,
%U A114160 20649996837971550,646340185330747425,21521124899877175650,759572031366463998375,28325808256035867711750,1112907316518036732317625
%N A114160 E.g.f. is A(x) = (1-log(B(x)))/B(x), where B(x) = sqrt(1-2*x).
%C A114160 From _John M. Campbell_, May 20 2011: (Start)
%C A114160 a(n) is the determinant of the n X n matrix of the form:
%C A114160 |2 1 1 1  ...   1  |
%C A114160 |1 4 1 1  ...   1  |
%C A114160 |1 1 6 1  ...   1  |
%C A114160 |1 1 1 8  ...   1  |
%C A114160 |...    ...     1  |
%C A114160 |1 1 1 1  2n-2  1  |
%C A114160 |1 1 1 1   1    2n |
%C A114160 See examples. (End)
%D A114160 C. Dement, Floretion Integer Sequences (work in progress)
%H A114160 G. C. Greubel, <a href="/A114160/b114160.txt">Table of n, a(n) for n = 0..400</a>
%F A114160 a(n) = A001147(n) + A004041(n-1) = 2^n*Gamma(n+1/2)/Pi^(1/2)*(1/2*Psi(n+1/2)+1/2*gamma+log(2)+1). - _Vladeta Jovovic_
%e A114160 From _John M. Campbell_, May 20 2011: (Start)
%e A114160 Det[{
%e A114160 {2,1,1,1,1,1},
%e A114160 {1,4,1,1,1,1},
%e A114160 {1,1,6,1,1,1},
%e A114160 {1,1,1,8,1,1},
%e A114160 {1,1,1,1,10,1},
%e A114160 {1,1,1,1,1,12}}] = 29919 = a(6), and
%e A114160 Det[{
%e A114160 {2,1,1,1,1,1,1},
%e A114160 {1,4,1,1,1,1,1},
%e A114160 {1,1,6,1,1,1,1},
%e A114160 {1,1,1,8,1,1,1},
%e A114160 {1,1,1,1,10,1,1},
%e A114160 {1,1,1,1,1,12,1},
%e A114160 {1,1,1,1,1,1,14}}] = 399342 = a(7).
%e A114160 (End)
%t A114160 Range[0, 18]! CoefficientList[ Series[(1 - Log[Sqrt[1 - 2x]])/Sqrt[(1 - 2x)], {x, 0, 18}], x] (* or *)
%t A114160 f[n_] := FullSimplify[ 2^(n-1)*Gamma[n + 1/2]/Sqrt[Pi]*(PolyGamma[n + 1/2] + EulerGamma + Log[4] + 2)]; Table[f[n], {n, 0, 18}] (* _Robert G. Wilson v_ *)
%t A114160 twox[x_, y_] := If[x == y, 2*x, 1]; a[n_] := Det[Array[twox[#1, #2] &, {n, n}]]; Join[{1}, Table[a[n], {n, 1, 10}]] (* _John M. Campbell_, May 20 2011 *)
%o A114160 (PARI) my(x='x + O('x^50)); Vec(serlaplace((1 - log(sqrt(1 - 2*x)))/sqrt(1 - 2*x))) \\ _G. C. Greubel_, Feb 08 2017
%Y A114160 Cf. A114161.
%K A114160 nonn
%O A114160 0,2
%A A114160 _Creighton Dement_, Nov 14 2005
%E A114160 E.g.f. given by _Vladeta Jovovic_
%E A114160 More terms from _Robert G. Wilson v_, Nov 15 2005