A227234 - OEIS

%I #5 Jul 14 2013 12:07:54

%S 1,3,5,9,19,43,109,297,793,2059,5382,14319,38897,110525,335225,

%T 1067923,3449922,11058922,35087589,110642516,346605981,1072833978,

%U 3270252617,9869924183,29933522269,92890564700,298225920323,987831491085,3330591758612,11254395868044,37691422431130,124450270430236

%N G.f.: Sum_{n>=1} x^n * (1+x)^prime(n).

%F  a(n) = Sum_{k=1..n} binomial(prime(k), n-k).

%e G.f.: A(x) = x + 3*x^2 + 5*x^3 + 9*x^4 + 19*x^5 + 43*x^6 + 109*x^7 + 297*x^8 +...

%e where

%e A(x) = x*(1+x)^2 + x^2*(1+x)^3 + x^3*(1+x)^5 + x^4*(1+x)^7 + x^5*(1+x)^11 + x^6*(1+x)^13 + x^7*(1+x)^17 + x^8*(1+x)^19 +...+ x^n*(1+x)^prime(n) +...

%o (PARI) {a(n)=polcoeff(sum(m=1, n, x^m*(1+x+x*O(x^n))^prime(m)), n)}

%o for(n=1, 40, print1(a(n), ", "))

%o (PARI) {a(n) = sum(k=1,n,binomial(prime(k),n-k))}

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A227235, A000040.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 14 2013