A181411 a(n) = Sum_{k=0..n} C(n,k)*sigma(n+k) for n>=1.

4, 18, 55, 150, 379, 915, 2146, 4934, 11080, 24833, 54476, 119091, 259432, 556700, 1195135, 2561094, 5428597, 11488866, 24350993, 51296325, 107427025, 225330244, 472762497, 985966379, 2049357779, 4267962522, 8887535983, 18431783744

Offset: 1

Author

Paul D. Hanna, Oct 19 2010

Keywords

• nonn

Examples

			L.g.f.: L(x) = 4*x + 18*x^2/2 + 55*x^3/3 + 150*x^4/4 + 379*x^5/5 +...
Exponentiation yields the g.f. of A181410:
exp(L(x)) = 1 + 4*x + 17*x^2 + 65*x^3 + 234*x^4 + 804*x^5 +...
The initial terms begin:
a(1) = 1*1 + 1*3 = 4;
a(2) = 1*3 + 2*4 + 1*7 = 18;
a(3) = 1*4 + 3*7 + 3*6 + 1*12 = 55;
a(4) = 1*7 + 4*6 + 6*12 + 4*8 + 1*15 = 150; ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..3000
• Vaclav Kotesovec, Plot of a(n)/(n*2^n) for n = 1..10000

Crossrefs

Cf. A181410.

Programs

Mathematica

Table[Sum[Binomial[n,k] * DivisorSigma[1,n+k],{k,0,n}],{n,1,30}] (* Vaclav Kotesovec, Oct 05 2020 *)

PARI

{a(n)=sum(k=0,n,binomial(m,k)*sigma(n+k))}

Formula

Equals the logarithmic derivative of A181410.

Conjecture: a(n) ~ c * n * 2^n, where c = Pi^2/4 = A091476. - Vaclav Kotesovec, Oct 05 2020