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%I #68 Sep 29 2023 03:55:36
%S 1,1,2,7,37,266,2431,27007,353522,5329837,90960751,1733584106,
%T 36496226977,841146804577,21065166341402,569600638022431,
%U 16539483668991901,513293594376771362,16955228098102446847,593946277027962411007,21992967478132711654106,858319677924203716921141
%N a(0) = a(1) = 1; thereafter a(n) = (2*n-3)*a(n-1) + a(n-2).
%C A variant of A001515, which is the main entry.
%C a(n) = number of increasing ordered trees on the vertex set [0,n] (counted by the double factorials A001147) in which n is the label on the leaf that terminates the leftmost path from the root. - _David Callan_, Aug 24 2011
%H G. C. Greubel, <a href="/A144301/b144301.txt">Table of n, a(n) for n = 0..400</a>
%H E. Grosswald, <a href="http://dx.doi.org/10.1002/zamm.19800601018">Bessel Polynomials</a>, Lecture Notes Math., Vol. 698, 1978.
%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="http://dx.doi.org/10.1016/j.aml.2012.02.009">On the Stirling numbers associated with the meromorphic Weyl algebra</a>, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
%H W. Mlotkowski and A. Romanowicz, <a href="http://www.math.uni.wroc.pl/~pms/files/33.2/Article/33.2.19.pdf">A family of sequences of binomial type</a>, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
%F a(n) = A001515(n-1) for n>= 1.
%F E.g.f.: A(x) = exp(1-sqrt(1-2*x)) satisfies A'(x) = A(x)/(1-sqrt(1-2*x)).
%F Hence a(n+1) = Sum_{k=0..n} ( a(n-k)*binomial(n,k)*(2*k)!/(k!*2^k) ).
%F A''(x) = (A'(x)/(1-2*x))*(1 + 1/sqrt(1-2*x)).
%F A''(x) = 2*x*A''(x) + A'(x) + A(x), which is equivalent to the recurrence in the definition.
%F a(n) = Sum_{k=0..n-1} binomial(n+k-1,2*k)*(2*k)!/(k!*2^k) ). [See Grosswald, p. 6, Eq. (8).]
%F a(n) ~ exp(1)*(2n-1)!/(n!*2^n) as n -> oo. [See Grosswald, p. 124]
%F G.f.: 1+x/U(0) where U(k)= 1 - x - x*(2*k+1)/(1 - x - 2*x*(k+1)/U(k+1)) ; (continued fraction, 2-step). - _Sergei N. Gladkovskii_, Oct 06 2012
%F G.f.: 1+x*(1-x)/U(0) where U(k)= 1 - 3*x + x^2 - x*4*k - x^2*(2*k+1)*(2*k+2)/U(k+1) ; (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Oct 06 2012
%F E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)*(1+sqrt(1-2*x))/E(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 23 2013
%F G.f.: conjecture: 1 + x*(1-x)/(1-3*x+x^2)*Q(0), where Q(k) = 1 - 2*(k+1)*(2*k+1)*x^2/(2*(k+1)*(2*k+1)*x^2 - (1 - 3*x + x^2 - 4*x*k)*(1 - 7*x + x^2 - 4*x*k)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 19 2013
%F a(1 - n) = a(n) for all n in Z. (a(n+1) + a(n+2))^2 = a(n)*a(n+2) + a(n+1)*a(n+3) for all integer n. - _Michael Somos_, Nov 22 2013
%F G.f.: 1 + x/(1-x)*T(0), where T(k) = 1 - x*(k+1)/( x*(k+1) - (1-x)^2/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 26 2013
%e G.f. = 1 + x + 2*x^2 + 7*x^3 + 37*x^4 + 266*x^5 + 2431*x^6 + 27007*x^7 + ...
%t a[n_]:= HypergeometricPFQ[{n, 1 - n}, {}, -1/2]; (* _Michael Somos_, Nov 22 2013 *)
%t a[n_]:= With[{m= If[n<1, -n, n-1]}, Sum[(m+k)!/((m-k)! k! 2^k), {k,0,m}]]; (* _Michael Somos_, Nov 22 2013 *)
%t RecurrenceTable[{a[0]==a[1]==1, a[n]==(2*n-3)*a[n-1] +a[n-2]}, a, {n, 25}] (* _Vincenzo Librandi_, Jul 23 2015 *)
%t nxt[{n_,a_,b_}]:={n+1,b,b(2n-1)+a}; NestList[nxt,{1,1,1},30][[All,2]] (* _Harvey P. Dale_, Nov 29 2022 *)
%o (PARI) {a(n) = my(m = if( n<1, -n, n-1)); sum( k=0, m, (m+k)! / (k! * (m-k)! * 2^k))}; /* _Michael Somos_, Nov 22 2013 */
%o (Magma) [1] cat [n le 1 select n+1 else (2*n-1)*Self(n) + Self(n-1): n in [0..20]]; // _Vincenzo Librandi_, Jul 23 2015
%o (SageMath)
%o def A144301(n): return int(n==0) + sum(binomial(n-1,k)*factorial(n+k-1)/(2^k*factorial(n-1)) for k in range(n))
%o [A144301(n) for n in range(31)] # _G. C. Greubel_, Sep 29 2023
%Y See A001515 for much more about this sequence.
%Y See A144498 for first differences.
%K nonn,easy
%O 0,3
%A _David Applegate_ and _N. J. A. Sloane_, Dec 07 2008
%E More terms from _Vincenzo Librandi_, Jul 23 2015
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