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A080893 Expansion of e.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.

%I A080893 #58 Feb 16 2025 08:32:48
%S A080893 1,1,3,19,193,2721,49171,1084483,28245729,848456353,28875761731,
%T A080893 1098127402131,46150226651233,2124008553358849,106246577894593683,
%U A080893 5739439214861417731,332993721039856822081,20651350143685984386753
%N A080893 Expansion of e.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.
%C A080893 Essentially the same as A001517: a(n+1) = A001517(n).
%C A080893 For k >= 2, the difference a(n+k) - a(n) is divisible by k. It follows that for each k, the sequence formed by taking a(n) modulo k is periodic with period dividing k. For example, modulo 10 the sequence becomes [1, 1, 3, 9, 3, 1, 1, 3, 9, 3, ...], a purely periodic sequence of period 5. Cf. A047974. - _Peter Bala_, Feb 11 2025
%H A080893 W. Mlotkowski and A. Romanowicz, <a href="http://www.math.uni.wroc.pl/~pms/publicationsArticle.php?nr=33.2&amp;nrA=19&amp;ppB=401&amp;ppE=408">A family of sequences of binomial type</a>, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
%H A080893 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A080893 E.g.f.: exp((1-sqrt(1-4*x))/2).
%F A080893 D-finite with recurrence: a(n+2) = 2*(2*n + 1)*a(n+1) + a(n).
%F A080893 Recurrence: y(n+1) = Sum_{k = 0..n} binomial(n, k)*binomial(2k, k)*k!*y(n-k).
%F A080893 a(1 - n) = a(n).  a(n + 1) = A001517(n). - _Michael Somos_, Apr 07 2012
%F A080893 G.f.: 1 + x/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 17 2013
%F A080893 a(n) ~ 2^(2*n-3/2)*n^(n-1)/exp(n-1/2). - _Vaclav Kotesovec_, Jun 26 2013
%F A080893 a(n) = hypergeom([-n+1, n], [], -1). - _Peter Luschny_, Oct 17 2014
%F A080893 a(n) = Sum_{k=0..n} (-4)^(n-k) * Stirling1(n,k) * A009235(k) = (-4)^n * Sum_{k=0..n} (1/2)^k * Stirling1(n,k) * Bell_k(-1/2), where Bell_n(x) is n-th Bell polynomial. - _Seiichi Manyama_, Aug 02 2024
%t A080893 y[x_] := y[x] = 2(2x - 3)y[x - 1] + y[x - 2]; y[0] = 1; y[1] = 1; Table[y[n],{n,0,17}]
%t A080893 With[{nn=20},CoefficientList[Series[Exp[(1-Sqrt[1-4x])/2],{x,0,nn}], x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 30 2011 *)
%o A080893 (PARI) {a(n) = if( n<1, n = 1 - n); n! * polcoeff( exp( (1 - sqrt(1 - 4*x + x * O(x^n))) / 2), n)} /* _Michael Somos_, Apr 07 2012 */
%o A080893 (Sage)
%o A080893 A080893 = lambda n: hypergeometric([-n+1, n], [], -1)
%o A080893 [simplify(A080893(n)) for n in (0..19)] # _Peter Luschny_, Oct 17 2014
%Y A080893 Cf. A000108, A001517, A009235, A375173.
%K A080893 easy,nice,nonn
%O A080893 0,3
%A A080893 _Emanuele Munarini_, Mar 31 2003