A181081 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n-2k*+1) * n/(n-k).

1, 3, 7, 39, 336, 4077, 68461, 1955295, 129385141, 17371664728, 3501431925168, 947675920795833, 355261887514210899, 236156938257380344851, 390707976511340699319417, 1324768245535417597286345871

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..90

Formula

L.g.f.: L(x) = Sum_{n>=1} ( Sum_{k=0..n} binomial(n,k)^(n-k+1)*x^k ) * x^n/n.

Logarithmic derivative of A181080.

Example

L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + ...
which equals the series:
log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^3*x + 3^2*x^2 + x^3)*x^3/3
+ (1+ 4^4*x + 6^3*x^2 + 4^2*x^3 + x^4)*x^4/4
+ (1+ 5^5*x + 10^4*x^2 + 10^3*x^3 + 5^2*x^4 + x^5)*x^5/5
+ (1+ 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181080:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 + 10641*x^7 + ...

Mathematica

Table[Sum[Binomial[n-k, k]^(n-2*k+1)*(n/(n-k)), {k, 0, Floor[n/2]}], {n, 20}] (* G. C. Greubel, Apr 04 2021 *)

Prog

(PARI) a(n)=sum(k=0, n\2, binomial(n-k, k)^(n-2*k+1)*n/(n-k))
(PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m, k)^(m-k+1)*x^k)*x^m/m)+x*O(x^n), n)}
(Magma) [(&+[Binomial(n-j, j)^(n-2*j+1)*(n/(n-j)): j in [0..Floor(n/2)]]): n in [1..20]]; // G. C. Greubel, Apr 04 2021
(Sage) [sum( binomial(n-k, k)^(n-2*k+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021

Crossrefs

Cf. A181080 (exp), variants: A181071, A166895.

Sequence in context: A307952 A074582 A105621 * A166895 A368627 A018998

Adjacent sequences: A181078 A181079 A181080 * A181082 A181083 A181084

Keyword

nonn

Author

Paul D. Hanna, Oct 02 2010

Status

approved