A222035 - OEIS

%I #10 Feb 10 2013 12:43:42

%S 1,1,3,21,283,6501,234363,12486181,939179643,96206801061,

%T 13036139824123,2281006244079141,505073418764455803,

%U 139105885712904768421,46950396347414793682683,19169208258193612244036901,9360235493546553882893232763,5411227090880585168975672572581

%N G.f.: Sum_{n>=0} Product_{k=1..n} (1 - 1/(1 + k*(k+1)*(k+2)/3!*x)).

%e G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 283*x^4 + 6501*x^5 + 234363*x^6 +...

%e where, by definition,

%e A(x) = 1 + 1*x/(1+x) + 1*4*x^2/((1+x)*(1+4*x)) + 1*4*10*x^3/((1+x)*(1+4*x)*(1+10*x)) + 1*4*10*20*x^4/((1+x)*(1+4*x)*(1+10*x)*(1+20*x)) + 1*4*10*20*35*x^5/((1+x)*(1+4*x)*(1+10*x)*(1+20*x)*(1+35*x)) + 1*4*10*20*35*56*x^6/((1+x)*(1+4*x)*(1+10*x)*(1+20*x)*(1+35*x)*(1+56*x)) +...

%o (PARI) {a(n)=local(A=1+x); for(i=1, 1, A=sum(m=0, n, prod(k=1, m,1-1/(1+k*(k+1)*(k+2)/3!*x +x*O(x^n))))); polcoeff(A,n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A000292 (tetrahedral numbers)

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 06 2013