%I #25 Nov 07 2023 19:47:29
%S 1,2,4,9,20,44,96,209,455,991,2159,4704,10249,22330,48651,105997,
%T 230938,503150,1096225,2388372,5203604,11337218,24700671,53815949,
%U 117250109,255455647,556567394,1212606837,2641935832,5756049469,12540844137
%N Sum of coefficients of polynomials defined in comments lines.
%C At the same time that I introduced the polynomials P(n,x) defined by P(0,x)=1 and for n>0, P(n,x) = (-1)^n/(n+1) + x*Sum_{ i=0..n-1 } ( (-1)^i/(i+1) )*P(n-1-i,x) (Gazette des Mathematiciens 1992), I gave the generalization P(0,x) = u(0), P(n,x) = u(n) + x*Sum_{ i=0..n-1 } u(i)*P(n-1-i,x).
%C For u(n), n>=0, = 1 1 1 2 3 4 5 6 7 8 ... the array of coefficients of the polynomials P(n,x) is:
%C 1
%C 1 1
%C 1 2 1
%C 2 3 3 1
%C 3 6 6 4 1
%C 4 11 13 10 5 1
%C 5 18 27 24 15 6 1
%C 6 28 51 55 40 21 7 1
%C whose row sums are the present sequence.
%C The alternating row sums are 1 0 0 1 0 0 0 -1 ...
%C The antidiagonal sums are 1 1 2 4 7 13 23 41 73 ...
%C The first column of the inverse matrix is 1 -1 1 -2 5 -11 25 -63 ...
%D Paul Curtz, Gazette des Mathématiciens, 1992, no. 52, p. 44.
%H Alois P. Heinz, <a href="/A129891/b129891.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (1-x+x^3)/(1-3*x+2*x^2-x^4). - _Alois P. Heinz_, Oct 14 2009
%p a:= n-> (Matrix([1, 1, 0, 1]). Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0, 1][i] else 0 fi)^n)[1, 1]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Oct 14 2009
%t u[n_ /; n < 3] = 1; u[n_] := n-1;
%t p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[ u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand;
%t row[n_] := CoefficientList[ p[n][x], x];
%t Table[row[n] // Total, {n, 0, 30}] (* _Jean-François Alcover_, Oct 02 2012 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^3)/(1-3*x+2*x^2-x^4) )); // _G. C. Greubel_, Oct 24 2023
%o (SageMath)
%o def A129891_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-x+x^3)/(1-3*x+2*x^2-x^4) ).list()
%o A129891_list(40) # _G. C. Greubel_, Oct 24 2023
%Y Sums of coefficients of polynomials defined in A140530.
%Y Cf. A129841, A129696, A130620.
%K nonn
%O 0,2
%A _Paul Curtz_, Jun 04 2007
%E Edited by _N. J. A. Sloane_, Jul 05 2007
%E More terms from _Alois P. Heinz_, Oct 14 2009