A181071 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(k+1) * n/(n-k).

1, 3, 7, 15, 66, 357, 1891, 20559, 257605, 3436908, 96199478, 2734569969, 96260508267, 6820892444439, 438665726703387, 43006289605790127, 7366025744010911808, 1099005822684238964181, 309398207716948885643749

Offset

1,2

Links

Harvey P. Dale, Table of n, a(n) for n = 1..122

Formula

Logarithmic derivative of A181070.

Example

L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 66*x^5/5 + ...
which equals the series:
L(x) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
+ (1+ 3^2*x + 3^3*x^2 + x^3)*x^3/3
+ (1+ 4^2*x + 6^3*x^2 + 4^4*x^3 + x^5)*x^4/4
+ (1+ 5^2*x + 10^3*x^2 + 10^4*x^3 + 5^5*x^4 + x^5)*x^5/5
+ (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 + ...
Exponentiation yields the g.f. of A181070:
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 + ...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A268061 A298358 A023370 * A258936 A336100 A096422

Adjacent sequences: A181068 A181069 A181070 * A181072 A181073 A181074

Keyword

nonn

Author

Paul D. Hanna, Oct 02 2010

Status

approved