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A141411 Defined in comments.

%I A141411 #22 Jan 31 2025 08:21:33
%S A141411 3,1,31,28,365,514,4388,8220,53871,122284,673222,1748055,8535397,
%T A141411 24383499,109449848,334783855,1415768769,4548229589,18434398665,
%U A141411 61345927764,241210652738,823296868656,3167642169823,11010462627756,41708741708554,146886286090602
%N A141411 Defined in comments.
%C A141411 Given any sequence {u(i), i >= 0} we define a family of polynomials by P(0,x) = u(0), P(n,x) = u(n) + x*Sum_{i=0..n-1} u(i)*P(n-i-1, x). Then we set a(n) = (P(n,-1)+P(n,1))/2.
%C A141411 For the present example we take {u(i)} to be 3,1,4,1,5,9,... (A000796).
%D A141411 P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
%D A141411 P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78 .
%H A141411 Alois P. Heinz, <a href="/A141411/b141411.txt">Table of n, a(n) for n = 0..500</a>
%F A141411 a(n) ~ c * d^n, where d = 3.6412947999106071671946396356753..., c = 1.387705266307957334035092183546... . - _Vaclav Kotesovec_, Sep 10 2014
%p A141411 u:= proc(n) Digits:= max(n+10);
%p A141411        trunc (10* frac(evalf(Pi*10^(n-1))))
%p A141411     end:
%p A141411 P:= proc(n) option remember; local i, x;
%p A141411       if n=0 then u(0)
%p A141411     else unapply(expand(u(n)+x*add(u(i)*P(n-i-1)(x), i=0..n-1)), x)
%p A141411       fi
%p A141411     end:
%p A141411 a:= n-> (P(n)(1)+P(n)(-1))/2:
%p A141411 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 06 2009
%t A141411 nmax = 25; digits = RealDigits[Pi, 10, nmax+1][[1]]; p[0][_] = digits[[1]]; p[n_][x_] := p[n][x] = digits[[n+1]] + x*Sum[digits[[i+1]] p[n-i-1][x], {i, 0, n-1}]; a[n_] := (p[n][1] + p[n][-1])/2; Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Nov 22 2012 *)
%Y A141411 See A130620 for another version.
%K A141411 nonn,easy
%O A141411 0,1
%A A141411 _Paul Curtz_, Jun 18 2007
%E A141411 Edited by _N. J. A. Sloane_, Aug 26 2009
%E A141411 Corrected and extended by _Alois P. Heinz_, Sep 06 2009