A301456
G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x)^k)^k.
Original entry on oeis.org
1, 1, 3, 12, 49, 217, 1006, 4810, 23576, 117812, 597937, 3073874, 15972678, 83758809, 442681653, 2355678968, 12610759255, 67868269712, 366979432955, 1992755590086, 10862329206524, 59414599714958, 326009477088080, 1793977307978268, 9898072238695390, 54744525395860053, 303463833091357785
Offset: 0
G.f. A(x) = 1 + x + 3*x^2 + 12*x^3 + 49*x^4 + 217*x^5 + 1006*x^6 + 4810*x^7 + 23576*x^8 + 117812*x^9 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * (1 + x^4*A(x)^4)^4 * ...
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 141*x^4/4 + 751*x^5/5 + 4064*x^6/6 + 22198*x^7/7 + 122381*x^8/8 + ... + A270922(n)*x^n/n + ...
A301624
G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.
Original entry on oeis.org
1, -1, -1, 4, 1, -17, -6, 118, -8, -876, 625, 5966, -7486, -41937, 75969, 306312, -768637, -2164992, 7487063, 14461466, -70259884, -89410774, 646971980, 459817892, -5861484630, -1128608133, 52082250637, -15894742662, -453574650852, 366848121166, 3866670213663, -5215687717614
Offset: 0
G.f. A(x) = 1 - x - x^2 + 4*x^3 + x^4 - 17*x^5 - 6*x^6 + 118*x^7 - 8*x^8 - 876*x^9 + 625*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * (1 - x^4*A(x)^4)^4 * ...
log(A(x)) = -x - 3*x^2/2 + 8*x^3/3 + 13*x^4/4 - 51*x^5/5 - 120*x^6/6 + 538*x^7/7 + 781*x^8/8 - 5419*x^9/9 - 3053*x^10/10 + ... + A281267(n)*x^n/n + ...
Cf.
A000219,
A006195,
A066398,
A073592,
A109085,
A181315,
A278428,
A281267,
A301455,
A301456,
A301625.
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with(numtheory):
Order := 33:
Gser := solve(series(x*exp(add(sigma[2](n)*x^n/n, n = 1..32)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..32); # Peter Bala, Feb 09 2020
A301577
G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - k*x^k*A(x)^k).
Original entry on oeis.org
1, 1, 4, 16, 75, 366, 1887, 10010, 54493, 302302, 1703599, 9723774, 56101292, 326640411, 1916800425, 11325242328, 67316128903, 402245682741, 2414978550718, 14560379165160, 88122911824659, 535188028077586, 3260549998701951, 19921639754064470, 122041156818328779
Offset: 0
G.f. A(x) = 1 + x + 4*x^2 + 16*x^3 + 75*x^4 + 366*x^5 + 1887*x^6 + 10010*x^7 + 54493*x^8 + 302302*x^9 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - 2*x^2*A(x)^2) * (1 - 3*x^3*A(x)^3) * ...).
log(A(x)) = x + 7*x^2/2 + 37*x^3/3 + 219*x^4/4 + 1276*x^5/5 + 7687*x^6/6 + 46551*x^7/7 + 285043*x^8/8 + ... + A297329(n)*x^n/n + ...
A302171
G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x))^k.
Original entry on oeis.org
1, 1, 4, 14, 54, 213, 880, 3724, 16143, 71227, 319067, 1447160, 6633530, 30682425, 143028870, 671293632, 3169572659, 15044993968, 71752624923, 343658572717, 1652266087698, 7971518032791, 38581202763318, 187269381724629, 911404238805468, 4446493502832481, 21742327471261176
Offset: 0
G.f. A(x) = 1 + x + 4*x^2 + 14*x^3 + 54*x^4 + 213*x^5 + 880*x^6 + 3724*x^7 + 16143*x^8 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x))^2 * (1 - x^3*A(x))^3 * ...).
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nmax = 30; A[] = 0; Do[A[x] = 1/Product[(1 - x^k*A[x])^k, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)
A301625
G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^k*A(x)^k)/(1 - x^k*A(x)^k))^k.
Original entry on oeis.org
1, 2, 10, 60, 398, 2820, 20892, 159868, 1253758, 10024070, 81400672, 669532924, 5566386324, 46701736772, 394910202608, 3362210548344, 28797181196766, 247955463799812, 2145088563952510, 18636002388075260, 162523319555310664, 1422259430668179592, 12485554521209720492, 109922263517662775292
Offset: 0
G.f. A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 398*x^4 + 2820*x^5 + 20892*x^6 + 159868*x^7 + 1253758*x^8 + ...
G.f. A(x) satisfies: A(x) = ((1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * ...)/((1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = 2*x + 16*x^2/2 + 128*x^3/3 + 1056*x^4/4 + 8952*x^5/5 + 77200*x^6/6 + 673948*x^7/7 + 5937792*x^8/8 + ... + A270924(n)*x^n/n + ...
A301831
G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 + x^k*A(x)^k)^k.
Original entry on oeis.org
1, -1, 0, 0, 6, -16, 16, -34, 217, -681, 1343, -3466, 13370, -42380, 109477, -312448, 1040248, -3267138, 9447529, -28367596, 90504001, -283611105, 861087913, -2654231074, 8386506600, -26359974392, 81902319183, -256179313766, 809890745232, -2557697524240, 8046530976599
Offset: 0
G.f. A(x) = 1 - x + 6*x^4 - 16*x^5 + 16*x^6 - 34*x^7 + 217*x^8 - 681*x^9 + 1343*x^10 - 3466*x^11 + ...
log(A(x)) = -x - x^2/2 - x^3/3 + 23*x^4/4 - 51*x^5/5 + 35*x^6/6 - 197*x^7/7 + ... + A281266(n)*x^n/n + ...
Showing 1-6 of 6 results.