A184513 L.g.f.: Sum_{n>=1} (x^n/n)/sqrt(1 - 2*(2*x)^n) = Sum_{n>=1} a(n)*x^n/n.

1, 5, 19, 89, 351, 1601, 6469, 28129, 116119, 491865, 2032317, 8519969, 35154029, 146022609, 601843209, 2485436161, 10218366631, 42036132185, 172427570701, 707155973729, 2894803671877, 11843754333361, 48394276165561, 197620176468097, 806190115015101, 3286819758296625

Offset

1,2

Comments

Logarithmic derivative of A184512.

Links

Table of n, a(n) for n=1..26.

Formula

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

Example

L.g.f.: L(x) = x + 5*x^2/2 + 19*x^3/3 + 89*x^4/4 + 351*x^5/5 + ...
The l.g.f. equals the series:
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) + ...
The g.f. of A184512 begins:
exp(L(x)) = 1 + x + 3*x^2 + 9*x^3 + 33*x^4 + 115*x^5 + 445*x^6 + ...

Mathematica

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

Prog

(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, 2^((d-1)*(n/d-1))*binomial(2*(d-1), d-1)*d))}
(PARI) {a(n)=n*polcoeff(sum(m=1, n, (x^m/m)/sqrt(1-2*(2*x)^m+x*O(x^n))), n)}

Crossrefs

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

Sequence in context: A323788 A351373 A154598 * A149801 A149802 A149803

Adjacent sequences: A184510 A184511 A184512 * A184514 A184515 A184516

Keyword

nonn

Author

Paul D. Hanna, Mar 18 2011

Extensions

a(24)-a(26) from Amiram Eldar, Aug 18 2023

Status

approved