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A201951 G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + k*x + x^2).

%I A201951 #16 Oct 31 2014 06:20:35
%S A201951 1,1,1,3,6,13,33,85,234,675,2032,6367,20677,69442,240529,857634,
%T A201951 3141970,11808611,45464065,179088744,720947705,2962994169,12420658682,
%U A201951 53061133078,230828047288,1021809688593,4599749893986,21043392417004,97784119963565,461277854065112
%N A201951 G.f.: Sum_{n>=0} x^n * Product_{k=0..n-1} (1 + k*x + x^2).
%C A201951 Equals the antidiagonal sums of irregular triangle A201949.
%H A201951 Vaclav Kotesovec, <a href="/A201951/b201951.txt">Table of n, a(n) for n = 0..560</a>
%F A201951 G.f.: A(x) = 1/(1 - x*(1+x^2)/(1+x*(1+x^2) - x*(1+x+x^2)/(1+x*(1+x+x^2) - x*(1+2*x+x^2)/(1+x*(1+2*x+x^2) - x*(1+3*x+x^2)/(1+x*(1+3*x+x^2) +...))))), a continued fraction.
%F A201951 G.f.: A(x) =1 + x*(1+x^2)/(G(0) - x*(1+x^2)) ; G(k)= k*x^2 + 1 + x + x^3 - x*(1+x+x^2+x*k)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 28 2011
%e A201951 G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 6*x^4 + 13*x^5 + 33*x^6 + 85*x^7 +...
%e A201951 where the g.f. equals the series:
%e A201951 A(x) = 1 + x*(1+x^2) + x^2*(1+x^2)*(1+x+x^2) + x^3*(1+x^2)*(1+x+x^2)*(1+2*x+x^2) + x^4*(1+x^2)*(1+x+x^2)*(1+2*x+x^2)*(1+3*x+x^2) +...
%o A201951 (PARI) {a(n)=sum(k=0,n,polcoeff(prod(j=0,n-k-1,1+j*x+x^2),k))}
%o A201951 (PARI) {a(n)=polcoeff(sum(m=0,n,x^m*prod(j=0,m-1,1+j*x+x^2))+x*O(x^n),n)}
%o A201951 (PARI) {a(n)=local(CF=x+x*O(x)); for(k=1, n, CF=x*(1+(n-k)*x+x^2)/(1+x*(1+(n-k)*x+x^2)-CF)); polcoeff(1/(1-CF), n, x)}
%Y A201951 Cf. A201949, A201950.
%K A201951 nonn
%O A201951 0,4
%A A201951 _Paul D. Hanna_, Dec 06 2011