This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056045 #60 Aug 15 2024 03:32:59
%S A056045 1,3,4,11,6,42,8,107,94,308,12,1718,14,3538,3474,14827,18,68172,20,
%T A056045 205316,117632,705686,24,3587174,53156,10400952,4689778,41321522,30,
%U A056045 185903342,32,611635179,193542210,2333606816,7049188,10422970784,38
%N A056045 a(n) = Sum_{d|n} binomial(n,d).
%H A056045 Seiichi Manyama, <a href="/A056045/b056045.txt">Table of n, a(n) for n = 1..3329</a> (terms 1..500 from T. D. Noe)
%H A056045 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%F A056045 L.g.f.: A(x) = Sum_{n>=1} log( G(x^n,n) ) where G(x,n) = 1 + x*G(x,n)^n. L.g.f. A(x) satisfies: exp(A(x)) = g.f. of A110448. - _Paul D. Hanna_, Nov 10 2007
%F A056045 a(n) = Sum_{k=1..A000005(n)} A007318(n, A027750(k)). - _Reinhard Zumkeller_, Aug 13 2013
%F A056045 a(n) = Sum_{k=1..n} binomial(n,gcd(n,k))/phi(n/gcd(n,k)) = Sum_{k=1..n} binomial(n,n/gcd(n,k))/phi(n/gcd(n,k)) where phi = A000010. - _Richard L. Ollerton_, Nov 08 2021
%F A056045 a(n) = n+1 iff n is prime. - _Bernard Schott_, Nov 30 2021
%e A056045 A(x) = log(1/(1-x) * G(x^2,2) * G(x^3,3) * G(x^4,4) * ...)
%e A056045 where the functions G(x,n) are g.f.s of well-known sequences:
%e A056045 G(x,2) = g.f. of A000108 = 1 + x*G(x,2)^2;
%e A056045 G(x,3) = g.f. of A001764 = 1 + x*G(x,3)^3;
%e A056045 G(x,4) = g.f. of A002293 = 1 + x*G(x,4)^4; etc.
%t A056045 f[n_] := Sum[ Binomial[n, d], {d, Divisors@ n}]; Array[f, 37] (* _Robert G. Wilson v_, Apr 23 2005 *)
%t A056045 Total[Binomial[#,Divisors[#]]]&/@Range[40] (* _Harvey P. Dale_, Dec 08 2018 *)
%o A056045 (PARI) {a(n)=n*polcoeff(sum(m=1,n,log(1/x*serreverse(x/(1+x^m +x*O(x^n))))),n)} /* _Paul D. Hanna_, Nov 10 2007 */
%o A056045 (PARI) {a(n)=sumdiv(n,d,binomial(n,d))} /* _Paul D. Hanna_, Nov 10 2007 */
%o A056045 (Haskell)
%o A056045 a056045 n = sum $ map (a007318 n) $ a027750_row n
%o A056045 -- _Reinhard Zumkeller_, Aug 13 2013
%o A056045 (Python)
%o A056045 from math import comb
%o A056045 from sympy import divisors
%o A056045 def A056045(n): return sum(comb(n,d) for d in divisors(n,generator=True)) # _Chai Wah Wu_, Jul 22 2024
%Y A056045 Cf. A110448 (exp(A(x))); A000108 (Catalan numbers), A001764, A002293, A174462.
%Y A056045 Cf. A000010 (comments on Dirichlet sum formulas).
%Y A056045 Cf. A308943 (similar, with Product).
%K A056045 nice,nonn
%O A056045 1,2
%A A056045 _Labos Elemer_, Jul 25 2000