A244004 - OEIS

%I #11 Jun 18 2014 04:20:56

%S 1,5,109,22265,45300601,865092893645,151339416167296549,

%T 240067372975786948895105,3437874756032842865408439504241,

%U 443629315988897178081546025182658298645,515464811455682924553846499988261249729439172189

%N a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 3^k)^k.

%F a(n) = Sum_{k=0..n} binomial(n,k) * (1 + 3^k)^(n-k) * 3^(k^2).

%F O.g.f.: Sum_{n>=0} (1 + 3^n)^n * x^n / (1-x)^(n+1).

%F O.g.f.: Sum_{n>=0} 3^(n^2) * x^n / (1 - (1+3^n)*x)^(n+1).

%F E.g.f.: exp(x) * Sum_{n>=0} (1 + 3^n)^n * x^n / n!.

%F a(n) ~ 3^(n^2). - _Vaclav Kotesovec_, Jun 18 2014

%e O.g.f.: A(x) = 1 + 5*x + 109*x^2 + 22265*x^3 + 45300601*x^4 +...

%e where the g.f. may be expressed by the series identity:

%e A(x) = 1/(1-x) + 4*x/(1-x)^2 + 10^2*x^2/(1-x)^3 + 28^3*x^3/(1-x)^4 + 82^4*x^4/(1-x)^5 + 244^5*x^5/(1-x)^6 + 730^6*x^6/(1-x)^7 +...

%e A(x) = 1/(1-2*x) + 3*x/(1-4*x)^2 + 3^4*x^2/(1-10*x)^3 + 3^9*x^3/(1-28*x)^4 + 3^16*x^4/(1-82*x)^5 + 3^25*x^5/(1-244*x)^6 + 3^36*x^6/(1-730*x)^7 +...

%e Illustration of initial terms:

%e a(0) = 1;

%e a(1) = 1 + (1+3);

%e a(2) = 1 + 2*(1+3) + (1+3^2)^2;

%e a(3) = 1 + 3*(1+3) + 3*(1+3^2)^2 + (1+3^3)^3;

%e a(4) = 1 + 4*(1+3) + 6*(1+3^2)^2 + 4*(1+3^3)^3 + (1+3^4)^4;

%e a(5) = 1 + 5*(1+3) + 10*(1+3^2)^2 + 10*(1+3^3)^3 + 5*(1+3^4)^4 + (1+3^5)^5; ...

%e Also, by a binomial identity we have

%e a(0) = 1;

%e a(1) = 2 + 3;

%e a(2) = 2^2 + 2*(1+3)*3 + 3^4;

%e a(3) = 2^3 + 3*(1+3)^2*3 + 3*(1+3^2)*3^4 + 3^9;

%e a(4) = 2^4 + 4*(1+3)^3*3 + 6*(1+3^2)^2*3^4 + 4*(1+3^3)*3^9 + 3^16;

%e a(5) = 2^5 + 5*(1+3)^4*3 + 10*(1+3^2)^3*3^4 + 10*(1+3^3)^2*3^9 + 5*(1+3^4)*3^16 + 3^25; ...

%t Table[Sum[Binomial[n,k]*(1+3^k)^k,{k,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jun 18 2014 *)

%o (PARI) {a(n)=sum(k=0, n, binomial(n, k)*(1+3^k)^k)}

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

%o (PARI) {a(n)=sum(k=0, n, binomial(n, k)*(1+3^k)^(n-k)*3^(k^2))}

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

%Y Cf. A243918, A202989.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 17 2014