A207643 a(n) = 1 + (n-1) + (n-1)*[n/2-1] + (n-1)*[n/2-1]*[n/3-1] + (n-1)*[n/2-1]*[n/3-1]*[n/4-1] +... for n>0 with a(0)=1, where [x] = floor(x).

1, 1, 2, 3, 7, 9, 26, 31, 71, 129, 262, 291, 1222, 1333, 2198, 5139, 11881, 12673, 39594, 41923, 117326, 251841, 354292, 371163, 1870453, 2598577, 3456926, 7103955, 16665859, 17283113, 72923314, 75437911, 165990152, 335534913, 422310802, 695765699, 3589651696

Offset: 0

Author

Paul D. Hanna, Feb 19 2012

Keywords

• nonn
• nice

Comments

Radius of convergence of g.f. A(x) is near 0.54783..., with A(1/2) = 7.2672875151872...

Compare the definition of a(n) to the trivial binomial sum:

2^(n-1) = 1 + (n-1) + (n-1)*(n/2-1) + (n-1)*(n/2-1)*(n/3-1) + (n-1)*(n/2-1)*(n/3-1)*(n/4-1) +...

Examples

			a(2) = 1 + 1 = 2; a(3) = 1 + 2 = 3;
a(4) = 1 + 3 + 3*[4/2-1] = 7;
a(5) = 1 + 4 + 4*[5/2-1] = 9;
a(6) = 1 + 5 + 5*[6/2-1] + 5*[6/2-1]*[6/3-1] = 26;
a(7) = 1 + 6 + 6*[7/2-1] + 6*[7/2-1]*[7/3-1] = 31;
a(8) = 1 + 7 + 7*[8/2-1] + 7*[8/2-1]*[8/3-1] + 7*[8/2-1]*[8/3-1]*[8/4-1] = 71; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A075885, A207645, A207644, A207646, A207647.

Programs

Mathematica

a[n_] := 1 + Sum[ Product[ Floor[(n-j)/j], {j, 1, k}], {k, 1, n/2}]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Mar 06 2013 *)

PARI

{a(n)=1+sum(k=1,n,prod(j=1,k,floor(n/j-1)))}
for(n=0,50,print1(a(n),", "))

PARI

a(n)=my(t=1);1+sum(k=1,n,t*=n\k-1) \\ Charles R Greathouse IV, Feb 20 2012

Formula

a(n) = 1 + Sum_{k=1..[n/2]} Product_{j=1..k} floor( (n-j) / j ).

Equals row sums of irregular triangle A207645.