A174461
G.f.: exp( Sum_{n>=1} A174462(n)*x^n/n ) where A174462(n) = Sum_{d|n} C(n,d)^2.
Original entry on oeis.org
1, 1, 3, 6, 21, 32, 174, 236, 1310, 2609, 12579, 18150, 150980, 198821, 1471346, 2645433, 17956158, 24534384, 234506155, 304507520, 2773986000, 4315363549, 36311714888, 47769153478, 500399410005, 637747787407, 6468558255893, 9142971548460, 88936892205131
Offset: 0
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{a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sumdiv(m,d,binomial(m,d)^2))+x*O(x^n)),n)}
A056045
a(n) = Sum_{d|n} binomial(n,d).
Original entry on oeis.org
1, 3, 4, 11, 6, 42, 8, 107, 94, 308, 12, 1718, 14, 3538, 3474, 14827, 18, 68172, 20, 205316, 117632, 705686, 24, 3587174, 53156, 10400952, 4689778, 41321522, 30, 185903342, 32, 611635179, 193542210, 2333606816, 7049188, 10422970784, 38
Offset: 1
A(x) = log(1/(1-x) * G(x^2,2) * G(x^3,3) * G(x^4,4) * ...)
where the functions G(x,n) are g.f.s of well-known sequences:
G(x,2) = g.f. of A000108 = 1 + x*G(x,2)^2;
G(x,3) = g.f. of A001764 = 1 + x*G(x,3)^3;
G(x,4) = g.f. of A002293 = 1 + x*G(x,4)^4; etc.
Cf.
A000010 (comments on Dirichlet sum formulas).
Cf.
A308943 (similar, with Product).
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a056045 n = sum $ map (a007318 n) $ a027750_row n
-- Reinhard Zumkeller, Aug 13 2013
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f[n_] := Sum[ Binomial[n, d], {d, Divisors@ n}]; Array[f, 37] (* Robert G. Wilson v, Apr 23 2005 *)
Total[Binomial[#,Divisors[#]]]&/@Range[40] (* Harvey P. Dale, Dec 08 2018 *)
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{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 */
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{a(n)=sumdiv(n,d,binomial(n,d))} /* Paul D. Hanna, Nov 10 2007 */
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from math import comb
from sympy import divisors
def A056045(n): return sum(comb(n,d) for d in divisors(n,generator=True)) # Chai Wah Wu, Jul 22 2024
A306842
a(n) = Sum_{d|n} binomial(n,d)*binomial(n,n/d).
Original entry on oeis.org
1, 4, 6, 44, 10, 612, 14, 3936, 7074, 22700, 22, 339792, 26, 624652, 2732760, 6401232, 34, 45174204, 38, 220441080, 309304842, 325909628, 46, 7330314960, 2822796950, 6760390052, 27417926304, 78814587656, 58, 548150764560, 62, 1352747882944, 2111872688706
Offset: 1
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a[n_] := DivisorSum[n, Binomial[n, #] * Binomial[n, n/#] &]; Array[a, 30] (* Amiram Eldar, Jun 13 2021 *)
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{a(n) = sumdiv(n, d, binomial(n, d)*binomial(n, n/d))}
A306843
a(n) = Sum_{d|n} binomial(n,d)^3.
Original entry on oeis.org
1, 9, 28, 281, 126, 11592, 344, 365465, 593434, 16095134, 1332, 921113624, 2198, 40424993884, 27175280778, 2137777203097, 4914, 121331143444050, 6860, 6310445825215406, 1572228697798262, 351047164202718608, 12168, 20174300460344963864, 149975199312626
Offset: 1
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a[n_] := DivisorSum[n, Binomial[n, #]^3 &]; Array[a, 25] (* Amiram Eldar, Jun 13 2021 *)
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{a(n) = sumdiv(n, d, binomial(n, d)^3)}
Showing 1-4 of 4 results.
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