A181085 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n+1) * n/(n-k).

1, 3, 25, 327, 6336, 513657, 142074241, 52903930911, 36806786795365, 148308705637730728, 1318954828711012426638, 15279013243159345043036553, 534104982404807772659968455891, 97749134742042348389685885848315523

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..60

Example

L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + ...
which equals the series:
L(x) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2
+ (1+ 3^5*x + 3^6*x^2 + x^3)*x^3/3
+ (1+ 4^6*x + 6^7*x^2 + 4^8*x^3 + x^4)*x^4/4
+ (1+ 5^7*x + 10^8*x^2 + 10^9*x^3 + 5^10*x^4 + x^5)*x^5/5
+ (1+ 6^8*x + 15^9*x^2 + 20^10*x^3 + 15^11*x^4 + 6^12*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181084:
exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...

Mathematica

Table[Sum[Binomial[n-k, k]^(n+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+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)}
(Sage) [sum( binomial(n-k, k)^(n+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # G. C. Greubel, Apr 04 2021
(Magma) [(&+[Binomial(n-j, j)^(n+1)*(n/(n-j)): j in [0..Floor(n/2)]]): j in [1..20]]; // G. C. Greubel, Apr 04 2021

Crossrefs

Variants: A166895, A181071, A181081, A181083.

Cf. A181084 (exp).

Sequence in context: A001907 A212722 A236268 * A143635 A246756 A023997

Adjacent sequences: A181082 A181083 A181084 * A181086 A181087 A181088

Keyword

nonn

Author

Paul D. Hanna, Oct 28 2010

Status

approved