A207649 - OEIS

%I #19 Feb 22 2018 20:25:53

%S 1,1,2,9,60,580,7380,119700,2387280,57211056,1615703040,52950688560,

%T 1988213895360,84622184412480,4045503794492160,215531680987022400,

%U 12709328457015590400,824497514996060505600,58531734588587835294720,4525412571041289046479360,379428698362674642409728000,34365923697307101211784908800,3350637193653655199937586944000

%N E.g.f.: Sum_{n>=0} 1/n! * Product_{k=1..n} ((1+x)^k - 1).

%H Vaclav Kotesovec, <a href="/A207649/b207649.txt">Table of n, a(n) for n = 0..300</a>

%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 60*x^4/4! + 580*x^5/5! +...

%e such that, by definition,

%e A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^2-1)/2! + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)/3! + ((1+x)-1)*((1+x)^2-1)*((1+x)^3-1)*((1+x)^4-1)/4! +...

%e or, more explicitly,

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

%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,1/m!*prod(k=1,m,(1+x)^k-1 +x*O(x^n)) ),n)}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A179525.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 19 2012

%E a(21)-a(22) corrected by _Andrew Howroyd_, Feb 22 2018