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A039619 Second column of Jabotinsky-triangle A038455 related to A006963.

%I A039619 #26 May 17 2015 15:33:54
%S A039619 1,9,107,1650,31594,725592,19471500,598482000,20742534576,
%T A039619 800575997760,34059828307680,1583808130195200,79925022369273600,
%U A039619 4350478314951982080,254086498336122950400,15849890120755311667200
%N A039619 Second column of Jabotinsky-triangle A038455 related to A006963.
%C A039619 Explicit formula using partitions of n into 2 parts: cf. Knuth's paper for f(n,2), n >= 2, formula with f(k) as given above.
%H A039619 D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, Mathematica J. 2.1 (1992), no. 4, 67-78.
%H A039619 D. E. Knuth, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.04.390">Problem 11832</a>,The American Mathematical Monthly, Vol. 122, No. 4 (April 2015), page 390.
%H A039619 David Callan, <a href="/A039619/a039619.pdf">An identity for A039619</a>.
%F A039619 a(n) = sum(binomial(n-1, j-1)*f(j)*f(n-j), j=1..n-1) with f(k) := A006963(k+1) = (2*k+1)!/k!, k >= 1.
%F A039619 E.g.f.: log((1-sqrt(1-4*x))/x/2)^2/2. - _Vladeta Jovovic_, May 02 2003
%F A039619 a(n) ~ log(2) * n^(n-1) * 2^(2*n-1/2) / exp(n). - _Vaclav Kotesovec_, Oct 07 2013
%F A039619 a(n) = (H(2*n-1) - H(n)) * (2*n-1)! / n! where H() is Harmonic numbers [Knuth 2015]. - _Michael Somos_, May 11 2015
%F A039619 a(n) = (n-1)!*sum(binomial(2n-1-k, n-1-k)/k,k=1..n-1) [Callan 2015]. - _David Callan_, May 17 2015
%e A039619 G.f. = x^2 + 9*x^3 + 107*x^4 + 1650*x^5 + 31594*x^6 + 725592*x^7 + ...
%t A039619 Rest[Rest[CoefficientList[Series[Log[(1-Sqrt[1-4*x])/x/2]^2/2, {x, 0, 20}], x]* Range[0, 20]!]] (* _Vaclav Kotesovec_, Oct 07 2013 *)
%t A039619 a[ n_] := If[ n < 1, 0, Sum[ 1/k, {k, n + 1, 2 n - 1}] (2 n - 1)!/n!]; (* _Michael Somos_, May 11 2015 *)
%t A039619 a[ n_] := If[ n < 1, 0, Sum[ Times @@ Drop[ Range[ n - 1] + n, {k}], {k, n - 1}]]; (* _Michael Somos_, May 11 2015 *)
%o A039619 (PARI) {a(n) = if( n<1, 0, sum(k=n+1, 2*n-1, 1/k) * (2*n-1)! / n!)}; /* _Michael Somos_, May 11 2015 */
%Y A039619 Cf. A006963, A038455.
%Y A039619 Cf. A039646.
%K A039619 nonn
%O A039619 2,2
%A A039619 _Wolfdieter Lang_