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 A306810 Inverse binomial transform of the continued fraction expansion of e. 0
 2, -1, 2, -4, 7, -8, -2, 41, -134, 296, -485, 512, 82, -2107, 6562, -13852, 21871, -22600, -2186, 83105, -255878, 531440, -826685, 846368, 59050, -2952451, 9034498, -18600436, 28697815, -29229256, -1594322, 98848025, -301327046, 617003000, -947027861, 961376768, 43046722 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS Jackson Earles, Aaron Li, Adam Nelson, Marlo Terr, Sarah Arpin, and Ilia Mishev Binomial Transforms of Sequences, CU Boulder Experimental Math Lab, Spring 2019. FORMULA a(n) = Sum{k=0...n}(-1)^(n+k)*binomial(n,k)*b(k), where b(k) is the k-th term of the continued fraction expansion of e. Conjectures from Colin Barker, Mar 12 2019: (Start) G.f.: (2 + 13*x + 37*x^2 + 55*x^3 + 42*x^4 + 14*x^5 + 2*x^6) / ((1 + x)*(1 + 3*x + 3*x^2)^2). a(n) = - 7*a(n-1) - 21*a(n-2) - 33*a(n-3) - 27*a(n-4) - 9*a(n-5) for n>6. (End) EXAMPLE For n = 3, a(3) = -binomial(3,0)*2 + binomial(3,1)*1 - binomial(3,2)*2 + binomial(3,3)*1 = -4. MATHEMATICA nmax = 50; A003417 = ContinuedFraction[E, nmax+1]; Table[Sum[(-1)^(n + k)*Binomial[n, k]*A003417[[k + 1]], {k, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Apr 23 2020 *) PROG (Sage) def OEISInverse(N, seq):     BT = [seq[0]]     k = 1     while k< N:         next = 0         j = 0         while j <=k:             next = next + (((-1)^(j+k))*(binomial(k, j))*seq[j])             j = j+1         BT.append(next)         k = k+1     return BT econt = oeis('A003417') OEISInverse(50, econt) CROSSREFS Continued fraction of e: A003417. Binomial transform of continued fraction of e: A306809. Sequence in context: A058553 A038067 A136102 * A325747 A325672 A279586 Adjacent sequences:  A306807 A306808 A306809 * A306811 A306812 A306813 KEYWORD cofr,easy,sign AUTHOR Sarah Arpin, Mar 11 2019 STATUS approved

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Last modified June 22 17:18 EDT 2021. Contains 345388 sequences. (Running on oeis4.)