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A243107 Number of terms in a bordered skew determinant.

%I A243107 #34 Nov 14 2020 07:01:57
%S A243107 1,1,2,4,13,41,226,1072,9059,58123,657766,5268836,73980787,707506879,
%T A243107 11823958238,131277234376,2542107619081,32122718085497,
%U A243107 706963537444114,10015472595953908,246853433179370621,3874536631479770761,105709617658879558402
%N A243107 Number of terms in a bordered skew determinant.
%C A243107 Possibly a different attempt to count the same bordered skew determinants as in A002772.
%H A243107 Alois P. Heinz, <a href="/A243107/b243107.txt">Table of n, a(n) for n = 0..450</a>
%H A243107 T. Muir, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM9350.0003.001">The Theory of Determinants in the Historical Order of Development</a>, 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
%H A243107 J. J. Sylvester, <a href="http://www.jstor.org/stable/2369200">Note on determinants and duadic disynthemes</a>, American J of Math, ii, (1879), 89-90, 214-222.
%F A243107 a(n) = Sum_{k=0..floor(n/2)} binomial(n, n - 2*k) * A002370(k).
%F A243107 E.g.f.: exp(x+x^2/4) / (1-x^2)^(1/4).
%F A243107 a(n) ~ n! * GAMMA(3/4) * (exp(5/4) + (-1)^n * exp(-3/4)) / (Pi * 2^(3/4)* n^(3/4)). - _Vaclav Kotesovec_, Aug 20 2014
%p A243107 a:= proc(n) option remember; `if`(n<4, [1$2, 2, 4][n+1],
%p A243107       (2*a(n-1)+2*(n-1)^2*a(n-2)-2*(n-1)*(n-2)*a(n-3)
%p A243107        -(n-1)*(n-2)*(n-3)*a(n-4))/2)
%p A243107     end:
%p A243107 seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 19 2014
%t A243107 b[n_] := Gamma[n+1/2] HypergeometricPFQ[{1/4, -n}, {}, -4]/Sqrt[Pi];
%t A243107 a[n_] := Sum[Binomial[n, n-2k] b[k], {k, 0, n/2}];
%t A243107 a /@ Range[0, 30]
%t A243107 (* Second program: *)
%t A243107 a[n_] := a[n] = If[n < 4, {1, 1, 2, 4}[[n+1]], (2a[n-1] + 2(n-1)^2 a[n-2] - 2(n-1)(n-2)a[n-3] - (n-1)(n-2)(n-3) a[n-4])/2];
%t A243107 a /@ Range[0, 30] (* _Jean-François Alcover_, Nov 13 2020, after _Alois P. Heinz_ *)
%o A243107 (PARI) my(x='x+O('x^66)); Vec(serlaplace(exp(x+x^2/4) / (1-x^2)^(1/4))) \\ _Joerg Arndt_, Aug 20 2014
%Y A243107 Cf. A002771, A002772.
%K A243107 nonn
%O A243107 0,3
%A A243107 _Sean A. Irvine_, Aug 19 2014
%E A243107 a(0)=1 prepended by _Joerg Arndt_, Aug 24 2014