A180305
G.f.: 1/(1 + x*d/dx log(eta(x))), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
Original entry on oeis.org
1, 1, 4, 11, 34, 96, 288, 833, 2456, 7175, 21054, 61633, 180674, 529220, 1550800, 4543446, 13312552, 39004278, 114281748, 334837511, 981059294, 2874447292, 8421986238, 24675963950, 72299290794, 211833080161, 620659794584, 1818500391218, 5328110328116, 15611082044176, 45739647180588, 134014753120706, 392656158141832
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 11*x^3 + 34*x^4 + 96*x^5 + 288*x^6 +...
eta(x)^3/A(x) = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 + 104*x^21 +...+ A184363(n)*x^n +...
1 + x*d/dx log(eta(x)) = 1 - x - 3*x^2 - 4*x^3 - 7*x^4 - 6*x^5 - 12*x^6 - 8*x^7 - 15*x^8 +...+ -sigma(n)*x^n +...
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a:= proc(n) option remember; `if`(n=0, 1,
add(a(n-i)*numtheory[sigma](i), i=1..n))
end:
seq(a(n), n=0..36); # Alois P. Heinz, Feb 11 2021
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nmax = 50; CoefficientList[Series[1/(1 - Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 06 2017 *)
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{a(n)=polcoeff(1/(1+x*deriv(log(eta(x+x*O(x^n))))), n)}
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{a(n)=if(n==0,1,sum(k=0,n-1,sigma(n-k)*a(k)))}
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N=66; x='x+O('x^N); Vec(1/(1 - sum(k=1,N, x^k/(1-x^k)^2))) \\ Joerg Arndt, Mar 09 2014
A184366
G.f.: eta(x)^3*(1 - x*eta'(x)/eta(x)), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
Original entry on oeis.org
1, -2, 0, 0, 0, 0, 7, 0, 0, 0, -21, 0, 0, 0, 0, 44, 0, 0, 0, 0, 0, -78, 0, 0, 0, 0, 0, 0, 125, 0, 0, 0, 0, 0, 0, 0, -187, 0, 0, 0, 0, 0, 0, 0, 0, 266, 0, 0, 0, 0, 0, 0, 0, 0, 0, -364, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 483, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -625, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 792, 0, 0
Offset: 0
G.f.: A(x) = 1 - 2*x + 7*x^6 - 21*x^10 + 44*x^15 - 78*x^21 +...
A(x) = eta(x)^3*[1 - x*d/dx log(eta(x))] where
eta(x)^3 = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 +...+ (-1)^n*(2n+1)*x^(n(n+1)/2) +...
1 - x*d/dx log(eta(x)) = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 15*x^8 +...+ sigma(n)*x^n +...
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{a(n)=polcoeff(sum(m=0,n,-(-1)^m*(m-2)*(m+3)*(2*m+1)/6*x^(m*(m+1)/2)),n)}
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{a(n)=polcoeff(eta(x+x*O(x^n))^3*(1-x*deriv(log(eta(x+x*O(x^n))))),n)}
A184362
G.f.: eta(x) + x*eta'(x).
Original entry on oeis.org
1, -2, -3, 0, 0, 6, 0, 8, 0, 0, 0, 0, -13, 0, 0, -16, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0, 0, -36, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -71, 0, 0, 0, 0, 0, 0, -78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 93, 0, 0, 0
Offset: 0
G.f.: A(x) = 1 - 2*x - 3*x^2 + 6*x^5 + 8*x^7 - 13*x^12 - 16*x^15 + 23*x^22 + 27*x^26 - 36*x^35 - 41*x^40 +...
Illustrate the property: [x^n] A(x)/eta(x)^(n+1) = 0
in the table of coefficients of A(x)/eta(x)^(n+1) for n=0..10:
[1, -1, -3, -4, -7, -6, -12, -8, -15, -13, -18,...,-sigma(n),...];
[1,(0), -2, -6, -15, -28, -55, -90, -154, -240, -378,...];
[1, 1,(0), -5, -20, -54, -130, -275, -555, -1050, -1924,...];
[1, 2, 3,(0), -17, -72, -221, -572, -1350, -2958, -6160,...];
[1, 3, 7, 10,(0), -63, -287, -930, -2580, -6475, -15162,...];
[1, 4, 12, 26, 38,(0), -253, -1196, -4059, -11780, -31027,...];
[1, 5, 18, 49, 105, 153,(0), -1062, -5175, -18140, -54544,...];
[1, 6, 25, 80, 210, 442, 646,(0), -4615, -22990, -82671,...];
[1, 7, 33, 120, 363, 924, 1926, 2816,(0), -20570, -104285,...];
[1, 8, 42, 170, 575, 1668, 4161, 8602, 12585,(0), -93538,...];
[1, 9, 52, 231, 858, 2756, 7766, 19071, 39182, 57343,(0),...]; ...
so that the coefficient of x^n in A(x)/eta(x)^(n+1) is zero for n>=1.
Note: the g.f.s of the diagonals in the above table are powers of G(x),
where G(x) = 1/eta(x*G(x)) is the g.f. of A109085.
The g.f. of A184363 equals:
A(x)*eta(x)^2 = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 +...+ (-1)^n*(2n+1)*(n^2+n+6)/6*x^(n(n+1)/2) +...
Showing 1-3 of 3 results.
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