A157019 a(n) = Sum_{d|n} binomial(n/d+d-2, d-1).

1, 2, 2, 4, 2, 8, 2, 10, 8, 12, 2, 34, 2, 16, 32, 38, 2, 62, 2, 92, 58, 24, 2, 210, 72, 28, 92, 198, 2, 394, 2, 274, 134, 36, 422, 776, 2, 40, 184, 1142, 2, 1178, 2, 618, 1232, 48, 2, 2634, 926, 1482, 308, 964, 2, 2972, 2004, 4610, 382, 60, 2, 8576, 2, 64, 6470, 5130

Offset: 1

Author

R. J. Mathar, Feb 21 2009

Keywords

• easy
• nonn

Comments

Equals row sums of triangle A156348. - Gary W. Adamson & Mats Granvik, Feb 21 2009

a(n) = 2 iff n is prime.

The binomial transform (note the offset) is 0, 1, 4, 11, 28, 67, 156, 359, 818, 1847, 4146, 9275, ... - R. J. Mathar, Mar 03 2013

a(n) is the number of distinct paths that connect the starting (1,1) point to the hyperbola with equation (x * y = n), when the choice for a move is constrained to belong to { (x := x + 1), (y := y + 1) }. - Luc Rousseau, Jun 27 2017

Examples

			a(4) = 4 = 1 + 2 + 0 + 1.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paul D. Hanna)

Crossrefs

Cf. A081543, A018818, A156838 (Mobius transform).

Cf. A156348.

Cf. A000010.

Programs

Maple

A157019 := proc(n) add( binomial(n/d+d-2, d-1), d=numtheory[divisors](n) ) ; end:

Mathematica

a[n_] := Sum[Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

PARI

{a(n)=polcoeff(sum(m=1,n,x^m/(1-x^m+x*O(x^n))^m),n)} \\ Paul D. Hanna, Mar 01 2009

Formula

G.f.: A(x) = Sum_{n>=1} x^n/(1 - x^n)^n. - Paul D. Hanna, Mar 01 2009

a(n) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 2, gcd(n,k) - 1) / phi(n/gcd(n,k)) = Sum_{k=1..n} binomial(gcd(n,k) + n/gcd(n,k) - 2, n/gcd(n,k) - 1) / phi(n/gcd(n,k)) where phi = A000010. - Richard L. Ollerton, May 19 2021