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A007552 Exponentiation of Fibonacci numbers.

%I A007552 M2860 #37 Jul 08 2025 17:05:24
%S A007552 1,3,10,42,204,1127,6924,46704,342167,2700295,22799218,204799885,
%T A007552 1947993126,19540680497,206001380039,2275381566909,26261810071925,
%U A007552 315969045744894,3954454344433658,51382626410402336,691956435942841207
%N A007552 Exponentiation of Fibonacci numbers.
%D A007552 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007552 Alois P. Heinz, <a href="/A007552/b007552.txt">Table of n, a(n) for n = 1..500</a>
%H A007552 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002. [pLink to arXiv version]
%H A007552 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%F A007552 E.g.f.: exp(exp(x/2)*(sqrt(5)*cosh(x*sqrt(5)/2)+sinh(x*sqrt(5)/2))/sqrt(5)-1)-1. - _Vladimir Kruchinin_, Feb 27 2015
%p A007552 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
%t A007552 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_ *)
%o A007552 (PARI) Vec(serlaplace(exp( serconvol(Ser(1/(1-x-x^2)),exp(x))-1)))
%o A007552 /* ==> [1, 1, 3, 10, 42, 204, 1127, 6924, 46704,...] (note offset 0) */
%o A007552 /* _Joerg Arndt_, Jun 16 2010 */
%Y A007552 Cf. A006701, A256180.
%K A007552 nonn
%O A007552 1,2
%A A007552 _N. J. A. Sloane_