A101510 Diagonal sums of binomial-Möbius product.

1, 2, 5, 10, 21, 43, 87, 175, 352, 707, 1417, 2836, 5674, 11353, 22716, 45443, 90886, 181748, 363451, 726870, 1453773, 2907648, 5815315, 11630195, 23259059, 46515887, 93029852, 186060921, 372129424, 744272221, 1488552317, 2977079872

Offset: 0

Author

Paul Barry, Dec 05 2004

Keywords

• easy
• nonn

Comments

Diagonal sums of A101508.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Programs

Mathematica

a[n_] := Sum[If[Mod[i+1, k+1] == 0, Binomial[n-k, i], 0], {k, 0, n/2}, {i, 0, n-k}]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jan 24 2014 *)
nmax = 40; CoefficientList[Series[(1/x^2) * Sum[x*(x/(1-x))^k/(1-x*(x/(1-x))^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 18 2019 *)

PARI

a(n) = sum(k=0, n\2, sum(i=0, n-k, if (!Mod(i+1, k+1), binomial(n-k, i)))); \\ Michel Marcus, Mar 16 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} Sum_{i=0..n-k, (k+1)|(i+1)} binomial(n-k,i).

G.f.: (1/x^2) * Sum_{n>=1} a*z^n/(1-a*z^n) (generalized Lambert series) where z=x/(1-x) and a=x. - Joerg Arndt, Jan 30 2011

a(n) ~ log(2) * 2^(n+1). - Vaclav Kotesovec, Mar 18 2019