A171800 - OEIS

%I #8 Jan 20 2017 02:49:31

%S 1,5,65,2673,397953,228882753,520970490625,4723480504289025,

%T 170687922720157732865,24563695027660686202250241,

%U 14068441356460459384918212890625,32058887942708146080692278858371608577,290694663888102785007861162394348756446314497

%N a(n) = ((n+1)*2^n + 1)*(2^n + 1)^(n-1).

%H Vincenzo Librandi, <a href="/A171800/b171800.txt">Table of n, a(n) for n = 0..58</a>

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

%F E.g.f.: Sum_{n>=0} (n+1)*2^(n^2) * exp(2^n*x) * x^n/n!.

%e G.f.: A(x) = 1 + 5*x + 65*x^2 + 2673*x^3 + 397953*x^4 +...

%e A(x) = 1/(1-x) + 2*2*x/(1-2*x)^2 + 3*2^4*x^2/(1-2^2*x)^3 + 4*2^9*x^3/(1-2^3*x)^4 +...

%t Table[((n + 1)*2^n + 1)*(2^n + 1)^(n - 1), {n, 0, 15}] (* _Wesley Ivan Hurt_, Jan 19 2017 *)

%o (PARI) {a(n)=polcoeff(sum(m=0,n,(m+1)*2^(m^2)*x^m/(1-2^m*x+x*O(x^n))^(m+1)),n)}

%o (PARI) {a(n)=n!*polcoeff(sum(k=0, n, (k+1)*2^(k^2)*exp(2^k*x)*x^k/k!), n)}

%o (PARI) {a(n)=((n+1)*2^n+1)*(2^n+1)^(n-1)}

%Y Cf. A136516, A171801, A171799.

%K nonn,easy

%O 0,2

%A _Paul D. Hanna_, Jan 20 2010