A184513 - OEIS

%I #12 Aug 20 2023 10:50:36

%S 1,5,19,89,351,1601,6469,28129,116119,491865,2032317,8519969,35154029,

%T 146022609,601843209,2485436161,10218366631,42036132185,172427570701,

%U 707155973729,2894803671877,11843754333361,48394276165561,197620176468097,806190115015101,3286819758296625

%N L.g.f.: Sum_{n>=1} (x^n/n)/sqrt(1 - 2*(2*x)^n) = Sum_{n>=1} a(n)*x^n/n.

%C Logarithmic derivative of A184512.

%F a(n) = Sum_{d|n} 2^((d-1)*(n/d-1)) * A000984(d-1) * d  where A000984(n) = C(2n,n).

%e L.g.f.: L(x) = x + 5*x^2/2 + 19*x^3/3 + 89*x^4/4 + 351*x^5/5 + ...

%e The l.g.f. equals the series:

%e L(x) = x/sqrt(1-4*x) + (x^2/2)/sqrt(1-8*x^2) + (x^3/3)/sqrt(1-16*x^3) + (x^4/4)/sqrt(1-32*x^4) +  (x^5/5)/sqrt(1-64*x^5) + ...

%e The g.f. of A184512 begins:

%e exp(L(x)) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 115*x^5 + 445*x^6 + ...

%t a[n_] := DivisorSum[n, 2^((#-1)*(n/#-1)) * Binomial[2*(#-1), #-1] * # &]; Array[a, 25] (* _Amiram Eldar_, Aug 18 2023 *)

%o (PARI) {a(n)=if(n<1,0,sumdiv(n,d,2^((d-1)*(n/d-1))*binomial(2*(d-1),d-1)*d))}

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

%Y Cf. A184512 (exp), A000984 (central binomial coefficients).

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 18 2011

%E a(24)-a(26) from _Amiram Eldar_, Aug 18 2023