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%I #11 Aug 02 2023 07:17:29
%S 1,1,3,7,29,129,757,5185,41155,368351,3671635,40295943,482758111,
%T 6268066531,87668492115,1314023850727,21011431917453,357014074280785,
%U 6423561495057421,122004755658629081,2439367774898883497,51213663674167659301,1126452985959434543237
%N First column of the matrix power A173279(.,.)^j in the limit j->infinity.
%C We can generalize A173279 to other matrices derived from some sequence S by Smat(n,k) := S(n-r*k), r >= 2,
%C and find that they define sequences B(x) via S(x)= B(X)/B(x^r), b(n) = Sum_{t=0..n, n-t == 0 (mod r)} S(t)*B_{(n-t)/r}.
%C The sequence here is the case of S=A000142 and r=2.
%H Jean-François Alcover, <a href="/A173280/b173280.txt">Table of n, a(n) for n = 0..100</a>
%F A000142(x) = A(x)/A(x^2), where A(x) and A000142(x) are the o.g.f.'s associated with A000142 and this sequence here.
%F Sum_{n>=0} 1/a(n) = 2.519966353393413186683398448854995831308...
%F a(n) = (A173279^j)(n,0).
%F a(n) = Sum_{t=0..n, n-t even} t!*a_{(n-t)/2}. - _R. J. Mathar_, Feb 22 2010
%p A173280 := proc(n) option remember; local a,l; if n = 0 then 1; else a :=0 ; for l from n to 0 by -2 do a := a+ l!*procname((n-l)/2) : end do ; a ; end if; end proc:
%p seq(A173280(n),n=0..60) ; # _R. J. Mathar_, Feb 22 2010
%t A173280[n_] := A173280[n] = Module[{a, l}, If[n == 0, 1, a = 0; For[l = n, l >= 0, l -= 2, a = a + l!*A173280[(n-l)/2]]; a]];
%t Table[A173280[n], {n, 0, 100}] (* _Jean-François Alcover_, Aug 02 2023, after _R. J. Mathar_ *)
%Y Cf. A000142.
%K nonn
%O 0,3
%A _Gary W. Adamson_, Feb 14 2010
%E Extended, and invalid comment on convergence to e removed, by _R. J. Mathar_, Feb 22 2010
%E Index of B in the convolution formula in the comment corrected by _R. J. Mathar_, Mar 23 2010
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