[seqfan] Operational recurrences

[Olivier Gerard]

At 16:04 +0200 99.04.23, N. J. A. Sloane wrote:

->There is a very nice paper by Henry Gould in Vol 1 of Fib Quart.
->that defines a whole class of recurrences.  I just extracted
->one sequence from the paper, but
->many more could be obtained using the same
->approach.
->

I remember that two years ago, I made some experiments about
bizarre ways of obtaining known sequences with a leibnizian
frame of mind.

Example:

%I A000629
%S A000629 1,2,6,26,150,1082,9366,94586,1091670,14174522,204495126,
%T A000629 3245265146,56183135190,1053716696762,21282685940886,460566381955706,
%U A000629 10631309363962710,260741534058271802,6771069326513690646
%N A000629 Expansion of -ln(2 - e^x); also of exp(x)/(2-exp(x)).
%F A000629 a(n) = Sum {from k=1 to infinity} k^n/(2^k); a(n) = 1 + Sum
{from j=0 to n-1} C(n,j) a(j); number of combinations of a Simplex lock
having n buttons.
%D A000629 D. E. Knuth, personal communication.
%O A000629 0,2
%K A000629 nonn,easy
%A A000629 njas, dek, nsinger at eos.hitc.com (Nick Singer)
%t A000629 a[0]=1; a[n_]:=(a[n]=1+Sum[Binomial[n,k] a[n-k], {k,1,n}])
%D A000629 J D E Konhauser et al., Which Way Did the Bicycle Go?, MAA 1996,
p. 174.

Can be obtained by this process (quite similar to Gould)

u[1]=x
u[n+1] = F ( d/dx  ( G( u[n] ) ) )

where G is the transformation  x->Sin[x]
and F is the transformation    Cos[x]->1-x  and   Sin[x]->x


Olivier