A256180
Exponential transform of the Fibonacci numbers.
Original entry on oeis.org
1, 1, 2, 6, 21, 86, 404, 2121, 12264, 77272, 525941, 3839706, 29891370, 246906569, 2154904856, 19799299506, 190904273049, 1926229186162, 20288311652672, 222568337565537, 2537998989244956, 30029233006187756, 368050599579654557, 4665833729558724030
Offset: 0
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F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= proc(n) option remember; `if`(n=0, 1,
add(binomial(n-1, j-1) *F(j) *a(n-j), j=1..n))
end:
seq(a(n), n=0..30);
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Table[Sum[BellY[n, k, Fibonacci[Range[n]]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
A007552
Exponentiation of Fibonacci numbers.
Original entry on oeis.org
1, 3, 10, 42, 204, 1127, 6924, 46704, 342167, 2700295, 22799218, 204799885, 1947993126, 19540680497, 206001380039, 2275381566909, 26261810071925, 315969045744894, 3954454344433658, 51382626410402336, 691956435942841207
Offset: 1
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Alois P. Heinz, Table of n, a(n) for n = 1..500
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002. [pLink to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
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f:= proc(n) option remember; `if`(n<2, 1, f(n-1) +f(n-2)) end: a:= proc(n) option remember; f(n) +add(binomial(n-1, k-1) *f(k) *a(n-k), k=1..n-1) end: seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008
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f[n_] := f[n] = If[n<2, 1, f[n-1]+f[n-2]]; a[n_] := a[n] = f[n]+Sum [Binomial[n-1, k-1]*f[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
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Vec(serlaplace(exp( serconvol(Ser(1/(1-x-x^2)),exp(x))-1)))
/* ==> [1, 1, 3, 10, 42, 204, 1127, 6924, 46704,...] (note offset 0) */
/* Joerg Arndt, Jun 16 2010 */
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