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
A105862
a(n) = n * Sum_{d|n} binomial(n,d)/gcd(n,d).
Original entry on oeis.org
1, 5, 10, 29, 26, 122, 50, 317, 334, 830, 122, 4754, 170, 7698, 11510, 34237, 290, 159530, 362, 458054, 358592, 1413890, 530, 8236946, 266276, 20806102, 14087530, 85118762, 842, 404242022, 962, 1244530621, 580671266, 4667223134, 35896250
Offset: 1
L.g.f.: A(x) = x + 5/2*x^2 + 10/3*x^3 + 29/4*x^4 + 26/5*x^5 + 61/3*x^6 +...
L.g.f.: A(x) = LOG[1/(1-x) * G(x^2,2)^2 * G(x^3,3)^3 * G(x^4,4)^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.
Exponentiation of l.g.f. A(x) is expressed by a product that begins:
exp(A(x)) = [1 + x + x^2 + x^3 +...] * [1 + 2*x^2 + 5*x^4 + 14*x^6 +...] * [1 + 3*x^3 + 12*x^6 + 55*x^9 +...] * [1 + 4*x^4 + 22*x^8 + 140*x^12 +...] * ...
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f[n_] := Block[{d = Divisors[n]}, n*Plus @@ (Binomial[n, d]/GCD[n, d])]; Table[ f[n], {n, 35}]
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a(n)=n*polcoeff(sum(m=1,n,m*log(1/x*serreverse(x/(1+x^m +x*O(x^n))))),n)
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a(n)=if(n<1,0,n*sumdiv(n,d,binomial(n,d)/gcd(n,d))) \\ Paul D. Hanna, Nov 11 2007
A206290
G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/(1 + x^k) ).
Original entry on oeis.org
1, 1, 2, 3, 5, 7, 12, 17, 29, 44, 77, 114, 218, 330, 617, 987, 1913, 2968, 6068, 9500, 19263, 31399, 64268, 101702, 218891, 348559, 735823, 1205239, 2576727, 4119884, 9100854, 14588992, 31841260, 52163378, 114485092, 183947681, 414704366, 667453931, 1487920000
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 12*x^6 + 17*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n( x/(1 + x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + x^3 + x^4 + x^5 + x^6 +...+ x^(n+1) +...
G_2(x) = x + x^3 + 2*x^5 + 5*x^7 + 14*x^9 +...+ A000108(n)*x^(2*n+1) +...
G_3(x) = x + x^4 + 3*x^7 + 12*x^10 + 55*x^13 +...+ A001764(n)*x^(3*n+1) +...
G_4(x) = x + x^5 + 4*x^9 + 22*x^13 + 140*x^17 +...+ A002293(n)*x^(4*n+1) +...
G_5(x) = x + x^6 + 5*x^11 + 35*x^16 + 285*x^21 +...+ A002294(n)*x^(5*n+1) +...
G_6(x) = x + x^7 + 6*x^13 + 51*x^19 + 506*x^25 +...+ A002295(n)*x^(6*n+1) +...
G_7(x) = x + x^8 + 7*x^15 + 70*x^22 + 819*x^29 +...+ A002296(n)*x^(7*n+1) +...
Note that G_n(x) = x + x*G_n(x)^n.
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{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x/(1+x^k+x*O(x^n))))),n)}
for(n=0,45,print1(a(n),", "))
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)}
Showing 1-4 of 4 results.
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