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

0,2

Comments

Diagonal sums of A101508.

Links

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

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A027437 A267444 A267880 * A066819 A114279 A101400

Adjacent sequences: A101507 A101508 A101509 * A101511 A101512 A101513

Keyword

easy,nonn

Author

Paul Barry, Dec 05 2004

Status

approved