cp's OEIS Frontend

A306810 Inverse binomial transform of the continued fraction expansion of e.

%I A306810 #24 Aug 10 2024 16:22:16
%S A306810 2,-1,2,-4,7,-8,-2,41,-134,296,-485,512,82,-2107,6562,-13852,21871,
%T A306810 -22600,-2186,83105,-255878,531440,-826685,846368,59050,-2952451,
%U A306810 9034498,-18600436,28697815,-29229256,-1594322,98848025,-301327046,617003000,-947027861,961376768,43046722
%N A306810 Inverse binomial transform of the continued fraction expansion of e.
%H A306810 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 A306810 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 A306810 Conjectures from _Colin Barker_, Mar 12 2019: (Start)
%F A306810 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 A306810 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 A306810 (End)
%e A306810 For n = 3, a(3) = -binomial(3,0)*2 + binomial(3,1)*1 - binomial(3,2)*2 + binomial(3,3)*1 = -4.
%t A306810 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 A306810 (Sage)
%o A306810 def OEISInverse(N, seq):
%o A306810     BT = [seq[0]]
%o A306810     k = 1
%o A306810     while k< N:
%o A306810         next = 0
%o A306810         j = 0
%o A306810         while j <=k:
%o A306810             next = next + (((-1)^(j+k))*(binomial(k,j))*seq[j])
%o A306810             j = j+1
%o A306810         BT.append(next)
%o A306810         k = k+1
%o A306810     return BT
%o A306810 econt = oeis('A003417')
%o A306810 OEISInverse(50,econt)
%Y A306810 Continued fraction of e: A003417.
%Y A306810 Binomial transform of continued fraction of e: A306809.
%K A306810 easy,sign
%O A306810 0,1
%A A306810 _Sarah Arpin_, Mar 11 2019