A251684 - OEIS

%I #7 Aug 19 2024 15:06:01

%S 1,2,5,16,69,426,3947,55612,1177747,36816650,1676270109,110202314208,

%T 10408422663015,1407329003121294,271801891072128621,

%U 74846096423770137324,29351301902680241116593,16374214768286861089202358,12985582377076992552497257703,14629438237685095017820000611400

%N G.f.: exp( Sum_{n>=1} A047863(n)*x^n/n ), where A047863(n) = Sum_{k=0..n} binomial(n, k) * (2^k)^(n-k).

%C Logarithmic derivative yields A047863, the number of labeled graphs with 2-colored nodes where black nodes are only connected to white nodes and vice versa.

%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 16*x^3 + 69*x^4 + 426*x^5 + 3947*x^6 +...

%e where the logarithmic derivative yields A047863:

%e A'(x)/A(x) = 2 + 6*x + 26*x^2 + 162*x^3 + 1442*x^4 + 18306*x^5 + 330626*x^6 + 8488962*x^7 + 309465602*x^8 +...+ A047863(n+1)*x^n +...

%o (PARI) {A047863(n) = sum(k=0, n, binomial(n, k) * (2^k)^(n-k) )}

%o {a(n)=local(A);A=exp(sum(k=1,n+1, A047863(k)*x^k/k) +x*O(x^n)); polcoeff(A,n)}

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

%Y Cf. A047863.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 14 2015