|
%I #33 May 20 2021 10:55:36
%S 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,
%T 198,2,394,2,274,134,36,422,776,2,40,184,1142,2,1178,2,618,1232,48,2,
%U 2634,926,1482,308,964,2,2972,2004,4610,382,60,2,8576,2,64,6470,5130
%N a(n) = Sum_{d|n} binomial(n/d+d-2, d-1).
%C Equals row sums of triangle A156348. - _Gary W. Adamson_ & _Mats Granvik_, Feb 21 2009
%C a(n) = 2 iff n is prime.
%C 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
%C 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
%H Seiichi Manyama, <a href="/A157019/b157019.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Paul D. Hanna)
%F G.f.: A(x) = Sum_{n>=1} x^n/(1 - x^n)^n. - _Paul D. Hanna_, Mar 01 2009
%F 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
%e a(4) = 4 = 1 + 2 + 0 + 1.
%p A157019 := proc(n) add( binomial(n/d+d-2, d-1), d=numtheory[divisors](n) ) ; end:
%t a[n_] := Sum[Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}];
%t Array[a, 100] (* _Jean-François Alcover_, Mar 24 2020 *)
%o (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
%Y Cf. A081543, A018818, A156838 (Mobius transform).
%Y Cf. A156348.
%Y Cf. A000010.
%K easy,nonn
%O 1,2
%A _R. J. Mathar_, Feb 21 2009
|
