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 A091593 Reversion of Jacobsthal numbers A001045. 6
 1, -1, -1, 5, -3, -21, 51, 41, -391, 407, 1927, -6227, -2507, 49347, -71109, -236079, 966129, 9519, -7408497, 13685205, 32079981, -167077221, 60639939, 1209248505, -2761755543, -4457338681, 30629783831, -22124857219, -206064020315, 572040039283, 590258340811 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Hankel transform is (-2)^C(n+1,2). - Paul Barry, Apr 28 2009 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 FORMULA G.f.: (-(1+x)+sqrt(1+2*x+9*x^2))/(4*x^2). - Corrected by Seiichi Manyama, May 12 2019 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*C(k)*(-1)^(n-k)2^k, where C(n) is A000108. - Paul Barry, May 16 2005 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 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 Conjecture: (n+2)*a(n) +(2*n+1)*a(n-1) +9*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 26 2012 a(n) = (-1)^n*hypergeom([-n/2, (1-n)/2], [2], -8). - Peter Luschny, May 28 2014 R. J. Mathar's conjecture confirmed by Maple using this hypergeom form. - Robert Israel, Sep 22 2014 MAPLE a := n -> hypergeom([-n, -n-1], [2], -2); seq(round(evalf(a(n), 99)), n=0..30); # Peter Luschny, Sep 22 2014 MATHEMATICA 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 *) PROG (Maxima) a(n):=sum(2^(i)*(-1)^(n-i)*binomial(n+1, i)^2*(n-i+1)/(i+1), i, 0, n)/(n+1); (* Vladimir Kruchinin, Oct 12 2011 *) (Sage) # Algorithm of L. Seidel (1877) def A091593_list(n) :     D = [0]*(n+2); D[1] = 1     R = []; b = false; h = 1     for i in range(2*n) :         if b :             for k in range(1, h, 1) : D[k] += -2*D[k+1]             R.append(D[1])         else :             for k in range(h, 0, -1) : D[k] += D[k-1]             h += 1         b = not b     return R A091593_list(30)  # Peter Luschny, Oct 19 2012 CROSSREFS Cf. A154825. Sequence in context: A199638 A296356 A154825 * A139699 A303634 A069607 Adjacent sequences:  A091590 A091591 A091592 * A091594 A091595 A091596 KEYWORD easy,sign AUTHOR Paul Barry, Jan 23 2004 STATUS approved

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Last modified August 2 02:53 EDT 2021. Contains 346409 sequences. (Running on oeis4.)