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A130620 Defined in comments.

%I A130620 #27 Jan 27 2025 06:41:15
%S A130620 3,9,31,106,365,1263,4388,15336,53871,190059,673222,2393291,8535397,
%T A130620 30526712,109449848,393272258,1415768769,5105086517,18434398665,
%U A130620 66647658995,241210652738,873773659486,3167642169823,11491042716338,41708741708554,151461799255253
%N A130620 Defined in comments.
%C A130620 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).
%C A130620 Then a(n) is the sum of the odd coefficients of P(n,x) if n is odd and a(n) is the sum of the even coefficients otherwise: a(n) = ((-1)^n*P(n,-1) + P(n,1))/2.
%C A130620 For the present example we take {u(i)} to be 3,1,4,1,5,9,... (A000796).
%D A130620 P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
%D A130620 P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78 .
%H A130620 Alois P. Heinz, <a href="/A130620/b130620.txt">Table of n, a(n) for n = 0..500</a>
%F A130620 a(n) ~ c * d^n, where d = 3.6412947999106071671946396356753... (same as for A141411), c = 1.38770526630795733403509218... . - _Vaclav Kotesovec_, Sep 12 2014
%e A130620 We have P(0,x)=3, P(1,x)=1+9x, P(2,x)=4+6x+27x^2, ..., so that for example a(2) = (25+37)/2 = 31.
%e A130620 The polynomials P(n,x) are:
%e A130620 n=0: 3,
%e A130620 n=1: 1+ 9*x,
%e A130620 n=2: 4+ 6*x+ 27*x^2,
%e A130620 n=3: 1+25*x+ 27*x^2+ 81*x^3,
%e A130620 n=4: 5+14*x+117*x^2+108*x^3+243*x^4,
%e A130620 n=5: 9+48*x+100*x^2+486*x^3+405*x^4+729*x^5.
%p A130620 u:= proc(n) Digits:= max(n+10);
%p A130620        trunc(10* frac(evalf(Pi*10^(n-1))))
%p A130620     end:
%p A130620 P:= proc(n) option remember; local i, x;
%p A130620       if n=0 then u(0)
%p A130620     else unapply(expand(u(n)+x*add(u(i)*P(n-i-1)(x), i=0..n-1)), x)
%p A130620       fi
%p A130620     end:
%p A130620 a:= n-> (P(n)(1) +(-1)^n*P(n)(-1))/2:
%p A130620 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 06 2009
%t A130620 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] + (-1)^n*p[n][-1])/2; Table[a[n], {n, 0, nmax}] (* _Jean-François Alcover_, Nov 22 2012 *)
%Y A130620 See A141411 for another version.
%K A130620 nonn,base,easy
%O A130620 0,1
%A A130620 _Paul Curtz_, Jun 18 2007
%E A130620 Edited by _N. J. A. Sloane_, Aug 26 2009
%E A130620 Definition corrected and more terms from _Alois P. Heinz_, Sep 06 2009