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A091593 Reversion of Jacobsthal numbers A001045.

%I A091593 #51 Jun 25 2025 20:43:05
%S A091593 1,-1,-1,5,-3,-21,51,41,-391,407,1927,-6227,-2507,49347,-71109,
%T A091593 -236079,966129,9519,-7408497,13685205,32079981,-167077221,60639939,
%U A091593 1209248505,-2761755543,-4457338681,30629783831,-22124857219,-206064020315,572040039283,590258340811
%N A091593 Reversion of Jacobsthal numbers A001045.
%C A091593 Hankel transform is (-2)^C(n+1,2). - _Paul Barry_, Apr 28 2009
%H A091593 Seiichi Manyama, <a href="/A091593/b091593.txt">Table of n, a(n) for n = 0..1000</a>
%H A091593 <a href="/index/Res#revert">Index entries for reversions of series</a>
%F A091593 G.f.: (-(1+x)+sqrt(1+2*x+9*x^2))/(4*x^2). - Corrected by _Seiichi Manyama_, May 12 2019
%F A091593 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*C(k)*(-1)^(n-k)*2^k, where C(n) is A000108(n). - _Paul Barry_, May 16 2005
%F A091593 G.f.: 1/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+x+2x^2/(1+ ... (continued fraction). - _Paul Barry_, Apr 28 2009
%F A091593 a(n) = Sum_{i=0..n} (2^(i)*(-1)^(n-i)*binomial(n+1,i)^2*(n-i+1)/(i+1))/(n+1). - _Vladimir Kruchinin_, Oct 12 2011
%F A091593 Conjecture: (n+2)*a(n) +(2*n+1)*a(n-1) +9*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 26 2012
%F A091593 a(n) = (-1)^n*hypergeom([-n/2, (1-n)/2], [2], -8). - _Peter Luschny_, May 28 2014
%F A091593 R. J. Mathar's conjecture confirmed by Maple using this hypergeom form. - _Robert Israel_, Sep 22 2014
%F A091593 a(n) = Sum_{k = 0..n} (-2)^k * (1/(n+1))*binomial(n+1, k)*binomial(n+1, k+1) =  Sum_{k = 0..n} (-2)^k * N(n+1,k+1), where N(n,k) = A001263(n,k) are the Narayana numbers. - _Peter Bala_, Sep 01 2023
%p A091593 a := n -> hypergeom([-n,-n-1], [2], -2);
%p A091593 seq(simplify(a(n)), n=0..30); # _Peter Luschny_, Sep 22 2014
%p A091593 # Using function CompInv from A357588.
%p A091593 CompInv(25, n -> (2^n - (-1)^n)/3 ); # _Peter Luschny_, Oct 07 2022
%t A091593 a[n_] := Hypergeometric2F1[-n - 1, -n - 1, 2, -2] + (n + 1)*Hypergeometric2F1[-n, -n, 3, -2]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 03 2016, after _Vladimir Kruchinin_ *)
%o A091593 (Maxima)
%o A091593 a(n) := hypergeometric([ -n - 1, -n - 1 ], [ 2 ], -2) + (n + 1) * hypergeometric([ -n, -n ], [ 3 ], -2); /* _Vladimir Kruchinin_, Oct 12 2011 */
%o A091593 (Sage) # Algorithm of L. Seidel (1877)
%o A091593 def A091593_list(n) :
%o A091593     D = [0]*(n+2); D[1] = 1
%o A091593     R = []; b = false; h = 1
%o A091593     for i in range(2*n) :
%o A091593         if b :
%o A091593             for k in range(1, h, 1) : D[k] += -2*D[k+1]
%o A091593             R.append(D[1])
%o A091593         else :
%o A091593             for k in range(h, 0, -1) : D[k] += D[k-1]
%o A091593             h += 1
%o A091593         b = not b
%o A091593     return R
%o A091593 A091593_list(30)  # _Peter Luschny_, Oct 19 2012
%Y A091593 Cf. A001263, A154825.
%K A091593 easy,sign
%O A091593 0,4
%A A091593 _Paul Barry_, Jan 23 2004