cp's OEIS Frontend

A251184 a(n) = Sum_{k=0..n} binomial(n,k) * (2^k + 3)^k.

%I A251184 #9 Jan 25 2015 07:44:45
%S A251184 1,6,60,1494,135960,53187306,90775495620,662696938144254,
%T A251184 20254044105203565360,2548567387213968842106066,
%U A251184 1305307745923414524904985640540,2701621925224675918174411993192242534,22497571753504120612543889146545487252194120,751859379104473999264213367292907652977053019173306
%N A251184 a(n) = Sum_{k=0..n} binomial(n,k) * (2^k + 3)^k.
%F A251184 G.f.: Sum_{n>=0} (2^n + 3)^n * x^n / (1-x)^(n+1).
%F A251184 G.f.: Sum_{n>=0} 2^(n^2) * x^n / (1-x - 3*2^n*x)^(n+1).
%F A251184 a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Jan 25 2015
%e A251184 G.f.: A(x) = 1 + 6*x + 60*x^2 + 1494*x^3 + 135960*x^4 + 53187306*x^5 +...
%e A251184 where we have the identity:
%e A251184 (1) A(x) = 1/(1-x) + (2+3)*x/(1-x)^2 + (2^2+3)^2*x^2/(1-x)^3 + (2^3+3)^3*x^3/(1-x)^4 + (2^4+3)^4*x^4/(1-x)^5 + (2^5+3)^5*x^5/(1-x)^6 +...
%e A251184 (2) A(x) = 1/(1-4*x) + 2*x/(1-x - 3*2*x)^2 + 2^4*x^2/(1-x - 3*2^2*x)^3 + 2^9*x^3/(1-x - 3*2^3*x)^4 + 2^16*x^4/(1-x - 3*2^4*x)^5 + 2^25*x^5/(1-x - 3*2^5*x)^6 +...
%e A251184 Illustration of initial terms.
%e A251184 a(0) = 1;
%e A251184 a(1) = 1 + (2+3) = 6;
%e A251184 a(2) = 1 + 2*(2+3) + (2^2+3)^2 = 60;
%e A251184 a(3) = 1 + 3*(2+3) + 3*(2^2+3)^2 + (2^3+3)^3 = 1494;
%e A251184 a(4) = 1 + 4*(2+3) + 6*(2^2+3)^2 + 4*(2^3+3)^3 + (2^4+3)^4 = 135960;
%e A251184 a(5) = 1 + 5*(2+3) + 10*(2^2+3)^2 + 10*(2^3+3)^3 + 5*(2^4+3)^4 + (2^5+3)^5 = 53187306; ...
%o A251184 (PARI) {a(n)=sum(k=0, n, binomial(n, k) * (2^k + 3)^k )}
%o A251184 for(n=0, 15, print1(a(n), ", "))
%o A251184 (PARI) {a(n)=local(A=1); A=sum(m=0, n, (2^m + 3)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
%o A251184 for(n=0, 15, print1(a(n), ", "))
%o A251184 (PARI) {a(n)=local(A=1); A=sum(m=0, n, 2^(m^2) * x^m / (1-x - 3*2^m*x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
%o A251184 for(n=0, 15, print1(a(n), ", "))
%Y A251184 Cf. A251182, A251183.
%K A251184 nonn
%O A251184 0,2
%A A251184 _Paul D. Hanna_, Jan 20 2015