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%I #10 Sep 08 2022 08:45:53
%S 1,5,13,55,245,1215,6317,34187,190093,1079983,6239989,36554363,
%T 216600357,1295906671,7817665373,47499325915,290411653437,
%U 1785401003887,11030252590149,68444469966843,426386709191893,2665740642304879
%N a(n+1) = m + Sum_{j=0..n} (a(j)*a(n-j) + k) for n>=1, with a(0)=1, a(1)=5, k=1 and m=1.
%H G. C. Greubel, <a href="/A176648/b176648.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (1 - sqrt(1 - 4*z*(a(0) + (a(1) - a(0)^2)*z + (k+m)*z^2/(1-z) + k*z^2/(1-z)^2)))/(2*z) with a(0) = k = m = 1 and a(1) = 5.
%F (n+1)*a(n) - (7*n-2)*a(n-1) - (n-11)*a(n-2) + (23*n-70)*a(n-3) - 24*(n-4)*a(n-4) + 8*(n-5)*a(n-5) = 0. - _R. J. Mathar_, Feb 29 2016
%F From _G. C. Greubel_, Jul 02 2021: (Start)
%F a(n) = m + k*n + Sum_{j=0..n-1} a(j)*a(n-j-1) with a(0) = m = k = 1 and a(1) = 5.
%F G.f.: (1-x -sqrt(1 -6*x -7*x^2 +16*x^3 -8*x^4))/(2*x*(1-x)). (End)
%e a(2) = 2*1*5 + 2 + 1 = 13.
%e a(3) = 2*1*13 + 2 + 5^2 + 1 + 1 = 55.
%e a(4) = 2*1*55 + 2 + 2*5*13 + 2 + 1 = 245.
%p # First program
%p m:=1: : k := 1 : a(0):=1 : a(1):=5:
%p for n from 1 to 51 do a(n+1):=sum(a(p)*a(n-p)+k, p=0..n) +m : od :
%p seq(a(n), n=0..40);
%p # Second program
%p n:= 40;
%p S:= series((1-x -sqrt(1 -6*x -7*x^2 +16*x^3 -8*x^4))/(2*x*(1-x)), x, n+1);
%p seq(coeff(S, x, j), j = 0..n); # modified by _G. C. Greubel_, Jul 02 2021
%t a[n_,k_,m_]:= a[n,k,m]= If[n<2, 5^n, m +k*n +Sum[a[j,k,m]*a[n-j-1,k,m], {j,0,n-1}]];
%t Table[a[n, 1, 1], {n, 0, 40}] (* _G. C. Greubel_, Jul 02 2021 *)
%o (Magma)
%o R<x>:=PowerSeriesRing(Rationals(), 40);
%o Coefficients(R!( (1-x -Sqrt(1 -6*x -7*x^2 +16*x^3 -8*x^4))/(2*x*(1-x)) )); // _G. C. Greubel_, Jul 02 2021
%o (Sage)
%o @CachedFunction
%o def a(n,k,m): return 5^n if (n<2) else m + k*n + sum(a(j,k,m)*a(n-j-1,k,m) for j in (0..n-1))
%o [a(n,1,1) for n in (0..40)] # _G. C. Greubel_, Jul 02 2021
%Y Cf. A176612, A176645.
%K easy,nonn
%O 0,2
%A _Richard Choulet_, Apr 22 2010
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