A181071 - OEIS

%I #11 Apr 05 2021 00:06:25

%S 1,3,7,15,66,357,1891,20559,257605,3436908,96199478,2734569969,

%T 96260508267,6820892444439,438665726703387,43006289605790127,

%U 7366025744010911808,1099005822684238964181,309398207716948885643749

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

%H Harvey P. Dale, <a href="/A181071/b181071.txt">Table of n, a(n) for n = 1..122</a>

%F Logarithmic derivative of A181070.

%e L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 66*x^5/5 + ...

%e which equals the series:

%e L(x) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2

%e + (1+ 3^2*x + 3^3*x^2 + x^3)*x^3/3

%e + (1+ 4^2*x + 6^3*x^2 + 4^4*x^3 + x^5)*x^4/4

%e + (1+ 5^2*x + 10^3*x^2 + 10^4*x^3 + 5^5*x^4 + x^5)*x^5/5

%e + (1+ 6^2*x + 15^3*x^2 + 20^4*x^3 + 15^5*x^4 + 6^6*x^5 + x^6)*x^6/6 + ...

%e Exponentiation yields the g.f. of A181070:

%e exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 23*x^5 + 88*x^6 + 379*x^7 + 3044*x^8 + ...

%t Table[Sum[Binomial[n-k,k]^(k+1) n/(n-k),{k,0,Floor[n/2]}],{n,20}] (* _Harvey P. Dale_, Sep 25 2020 *)

%o (PARI) a(n)=sum(k=0, n\2, binomial(n-k, k)^(k+1)*n/(n-k))

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

%o (Magma) [(&+[Binomial(n-j,j)^(j+1)*(n/(n-j)): j in [0..Floor(n/2)]]): n in [1..20]]; // _G. C. Greubel_, Apr 04 2021

%o (Sage) [sum( binomial(n-k, k)^(k+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # _G. C. Greubel_, Apr 04 2021

%Y Cf. A181070 (exp), variants: A181073, A181081.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 02 2010