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 A306809 Binomial transform of the continued fraction expansion of e. 1
 2, 3, 6, 12, 23, 46, 98, 213, 458, 972, 2051, 4322, 9098, 19113, 40054, 83748, 174767, 364086, 757298, 1572861, 3262242, 6757500, 13981019, 28894090, 59652314, 123032913, 253522382, 521957844, 1073741831, 2207135966, 4533576578, 9305762469, 19088743546, 39131924268 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Taking this sequence as a continued fraction it seems to converge to 2.31601650488979... 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} 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 - 11*x + 27*x^2 - 41*x^3 + 40*x^4 - 22*x^5 + 6*x^6) / ((1 - x)*(1 - 2*x)^2*(1 - x + x^2)^2). a(n) = 7*a(n-1) - 21*a(n-2) + 37*a(n-3) - 43*a(n-4) + 33*a(n-5) - 16*a(n-6) + 4*a(n-7) for n>6. (End) EXAMPLE For n = 3, the a(3) = binomial(3,0)*2 + binomial(3,1)*1 + binomial(3,2)*2 + binomial(3,3)*1 = 12. MATHEMATICA nmax = 50; A003417 = ContinuedFraction[E, nmax+1]; Table[Sum[Binomial[n, k]*A003417[[k + 1]], {k, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Apr 23 2020 *) PROG (Sage) def OEISbinomial_transform(N, seq):     BT = [seq[0]]     k = 1     while k< N:         next = 0         j = 0         while j <=k:             next = next + ((binomial(k, j))*seq[j])             j = j+1         BT.append(next)         k = k+1     return BT econt = oeis('A003417') OEISbinomial_transform(50, econt) CROSSREFS Cf. A003417 (continued fraction for e). Sequence in context: A293363 A326021 A164363 * A103341 A023675 A029996 Adjacent sequences:  A306806 A306807 A306808 * A306810 A306811 A306812 KEYWORD cofr,nonn,easy AUTHOR Sarah Arpin, Mar 11 2019 STATUS approved

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Last modified May 7 15:02 EDT 2021. Contains 343650 sequences. (Running on oeis4.)