A158887 a(n) = (n+1)^n * n! * binomial(n-1 + 1/(n+1), n).

1, 1, 4, 45, 1056, 43225, 2756160, 253586025, 31872332800, 5252921480961, 1099886703552000, 285322741626047125, 89844523369696972800, 33764841634845724313625, 14930493174337400252809216

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..230

Formula

a(n) = Product_{k=0..n-1} (k*(n+1) + 1) for n>0 with a(0)=1.

a(n) = coefficient of x^n/(n!*(n+1)^n) in 1/(1-x)^(1/(n+1)).

a(n) ~ sqrt(2*Pi) * exp(1-n) * n^(2*n-3/2). - Vaclav Kotesovec, Jun 28 2015

a(n) = (1+n)^n * gamma(n+1/(n+1)) / gamma(1/(n+1)). - Gerry Martens, May 30 2018

Example

a(1) = 1, a(2) = 1*4, a(3) = 1*5*9, a(4) = 1*6*11*16, a(5) = 1*7*13*19*25.

Maple

seq(mul(j*(n+1)+1, j=0..n-1), n = 0..15); # G. C. Greubel, Mar 04 2020

Mathematica

Table[(n+1)^n n!Binomial[n-1+1/(n+1), n], {n, 0, 20}] (* Harvey P. Dale, Oct 26 2011 *)
a[n_] := (1 + n)^n Gamma[n + 1/(1 + n)]/Gamma[1/(n + 1)] // FullSimplify
Table[a[n], {n, 0, 20} (* Gerry Martens, May 30 2018 *)

Prog

(PARI) a(n)=(n+1)^n*n!*polcoeff(1/(1-x+x*O(x^n))^(1/(n+1)), n)
(PARI) a(n)=if(n==0, 1, prod(k=0, n-1, k*(n+1)+1))
(Magma) [1] cat [&*[j*(n+1)+1: j in [0..n-1]]: n in [1..15]]; // G. C. Greubel, Mar 04 2020
(Sage) [product(j*(n+1)+1 for j in (0..n-1)) for n in (0..15)] # G. C. Greubel, Mar 04 2020
(GAP) List([0..15], n-> Product([0..n-1], j-> j*(n+1)+1) ); # G. C. Greubel, Mar 04 2020

Crossrefs

Sequence in context: A338456 A276292 A174484 * A126452 A082765 A132873

Adjacent sequences: A158884 A158885 A158886 * A158888 A158889 A158890

Keyword

nonn

Author

Paul D. Hanna, May 01 2009

Status

approved