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 A075271 a(0) = 1 and, for n >= 1, (BM)a(n) = 2*a(n-1), where BM is the BinomialMean transform. 12
 1, 3, 17, 211, 5793, 339491, 41326513, 10282961907, 5181436229441, 5258784071302723, 10717167529963833681, 43779339268428732008723, 358114286723184561034838497, 5862685570087914880854259126371, 192026370558313054275618817346778353 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The BinomialMean transform BM is defined by (BM)a(n) = (M^n)a(0) where (M)a(n) is the mean (a(n) + a(n+1))/2, or, alternatively, by (BM)a(n) = (Sum_{k=0..n} binomial(n,k)*a(k))/(2^n). The BinomialMean transform of this sequence is given in A075272. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..50 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1. FORMULA O.g.f. as a continued fraction: A(x) = 1/(1 + x - 2^2*x/(1 - 2*(2 - 1)^2*x/(1 + x - 2^4*x/(1 - 2*(2^2 - 1)^2*x/(1 + x - 2^6*x/(1 - 2*(2^3 - 1)^2*x/(1 + x - 2^8*x/(1 - 2*(2^4 - 1)^2*x/(1 + x - ... ))))))))). Cf. A075272. - Peter Bala, Nov 10 2017 EXAMPLE Given that a(0)=1 and a(1)=3. Then (BM)a(2) = (1 + 2*3 + a(2))/4 = 2a(1) = 6, hence a(2)=17. MAPLE iBM:= proc(p) proc(n) option remember; add(2^(k)*p(k)*(-1)^(n-k) *binomial(n, k), k=0..n) end end: a:= iBM(aa): aa:= n-> `if`(n=0, 1, 2*a(n-1)): seq(a(n), n=0..16);  # Alois P. Heinz, Sep 09 2008 MATHEMATICA iBM[p_] := Module[{proc}, proc[n_] := proc[n] = Sum[2^k*p[k]*(-1)^(n-k) * Binomial[n, k], {k, 0, n}]; proc]; a = iBM[aa]; aa[n_] := If[n == 0, 1, 2*a[n-1]]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Nov 08 2015, after Alois P. Heinz *) Table[Sum[QFactorial[k, 2] Binomial[n + 1, k]/2, {k, 0, n + 1}], {n, 0, 15}] (* Vladimir Reshetnikov, Oct 16 2016 *) CROSSREFS Cf. A075272. Sequence in context: A210898 A009494 A267659 * A194925 A072350 A181032 Adjacent sequences:  A075268 A075269 A075270 * A075272 A075273 A075274 KEYWORD eigen,nonn AUTHOR John W. Layman, Sep 11 2002 EXTENSIONS More terms from Alois P. Heinz, Sep 09 2008 STATUS approved

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Last modified October 14 23:59 EDT 2019. Contains 328025 sequences. (Running on oeis4.)