A251658 - OEIS

%I #22 Aug 26 2017 19:06:05

%S 1,3,13,75,471,3273,26916,262027,2851375,33353433,422098711,

%T 5947420449,94264842933,1637995975164,29950444958268,558160521126939,

%U 10427042476673645,194220965933772135,3611256888560663173,67328916459842949045,1266687562134302300334,24238971105618591823339

%N a(n) = Sum_{k=1..n} C(n,k) * sigma(k)^(n-k) for n>=1.

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

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

%e E.g.f.: A(x) = x + 3*x^2/2! + 13*x^3/3! + 75*x^4/4! + 471*x^5/5! +...

%e where

%e A(x) = exp(x)*x + exp(3*x)*x^2/2! + exp(4*x)*x^3/3! + exp(7*x)*x^4/4! + exp(6*x)*x^5/5! + exp(12*x)*x^6/6! + exp(8*x)*x^7/7! + exp(15*x)*x^8/8! +...+ exp(sigma(n)*x)*x^n/n! +...

%e RELATED EXPRESSIONS.

%e O.g.f.: A(x) = x + 3*x^2 + 13*x^3 + 75*x^4 + 471*x^5 + 3273*x^6 +...

%e where

%e A(x) = x/(1-x)^2 + x^2/(1-3*x)^3 + x^3/(1-4*x)^4 + x^4/(1-7*x)^5 + x^5/(1-6*x)^6 + x^6/(1-12*x)^7 + x^7/(1-8*x)^8 + x^8/(1-15*x)^9 +...+ x^n/(1-sigma(n)*x)^(n+1) +...

%e L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 75*x^4/4 + 471*x^5/5 + 3273*x^6/6 + 26916*x^7/7 + 262027*x^8/8 + 2851375*x^9/9 +...

%e where exponentiation yields the g.f. of A291558:

%e exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 25*x^4 + 123*x^5 + 695*x^6 + 4710*x^7 + 38573*x^8 + 363290*x^9 + 3762248*x^10 +...+ A291558(n)*x^n +...

%e which is an integer series.

%t Table[Sum[Binomial[n,k] * DivisorSigma[1,k]^(n-k), {k,1,n}], {n,1,20}] (* _Vaclav Kotesovec_, Aug 22 2017 *)

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

%o for(n=1, 25, print1(a(n), ", "))

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

%o for(n=1, 25, print1(a(n), ", "))

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

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A291558 (exp).

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 05 2015