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%I #21 Apr 23 2020 04:12:23
%S 2,-1,2,-4,7,-8,-2,41,-134,296,-485,512,82,-2107,6562,-13852,21871,
%T -22600,-2186,83105,-255878,531440,-826685,846368,59050,-2952451,
%U 9034498,-18600436,28697815,-29229256,-1594322,98848025,-301327046,617003000,-947027861,961376768,43046722
%N Inverse binomial transform of the continued fraction expansion of e.
%H Jackson Earles, Aaron Li, Adam Nelson, Marlo Terr, Sarah Arpin, and Ilia Mishev <a href="https://www.colorado.edu/math/binomial-transforms-sequences-spring-2019">Binomial Transforms of Sequences</a>, CU Boulder Experimental Math Lab, Spring 2019.
%F 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.
%F Conjectures from _Colin Barker_, Mar 12 2019: (Start)
%F 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).
%F 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.
%F (End)
%e For n = 3, a(3) = -binomial(3,0)*2 + binomial(3,1)*1 - binomial(3,2)*2 + binomial(3,3)*1 = -4.
%t 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 *)
%o (Sage)
%o def OEISInverse(N, seq):
%o BT = [seq[0]]
%o k = 1
%o while k< N:
%o next = 0
%o j = 0
%o while j <=k:
%o next = next + (((-1)^(j+k))*(binomial(k,j))*seq[j])
%o j = j+1
%o BT.append(next)
%o k = k+1
%o return BT
%o econt = oeis('A003417')
%o OEISInverse(50,econt)
%Y Continued fraction of e: A003417.
%Y Binomial transform of continued fraction of e: A306809.
%K cofr,easy,sign
%O 0,1
%A _Sarah Arpin_, Mar 11 2019
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