A255526
Coefficient of x^n in Product_{k>=1} 1/(1+x^k)^n.
Original entry on oeis.org
-1, 1, -4, 17, -56, 172, -547, 1809, -6061, 20316, -68135, 229244, -774372, 2624119, -8912759, 30328593, -103382254, 352975681, -1206921212, 4132159452, -14163858895, 48601267199, -166930975524, 573872089212, -1974472043081, 6798561779868, -23425506369715
Offset: 1
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Table[SeriesCoefficient[Product[1/(1+x^k)^n,{k,1,n}],{x,0,n}],{n,1,30}]
(* Calculation of constant c: *) 1/Sqrt[(4 - r^2*s^3*Derivative[0, 2][QPochhammer][-1, r*s])*Pi] /. FindRoot[{QPochhammer[-1, r*s] == 2/s, 2/s + r*s*Derivative[0, 1][QPochhammer][-1, r*s] == 0}, {r, -1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)
A278428
Series reversion of g.f. (1/2)*x*(-1; -x)_inf, where (a; q)_inf is the q-Pochhammer symbol.
Original entry on oeis.org
1, 1, 1, 2, 6, 17, 46, 128, 373, 1119, 3405, 10464, 32478, 101781, 321642, 1023512, 3276326, 10543100, 34088806, 110690682, 360810160, 1180195810, 3872588051, 12743937024, 42049240694, 139082885503, 461072582522, 1531697761470, 5098246648103, 17000237006441
Offset: 1
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InverseSeries[x QPochhammer[-1, -x]/2 + O[x]^35][[3]]
(* Calculation of constant c: *) 1/Sqrt[(4/s^2 - s*Derivative[0, 2][QPochhammer][-1, -s]/r) * Pi] /. FindRoot[{2*r == s*QPochhammer[-1, -s], 2*r == s^2*Derivative[0, 1][QPochhammer][-1, -s]}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)
A303174
a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).
Original entry on oeis.org
1, -1, 2, -5, 18, -60, 189, -601, 1967, -6544, 21872, -73247, 246080, -829924, 2808357, -9527485, 32389671, -110316862, 376372802, -1286063899, 4400499380, -15075608840, 51704898623, -177513230200, 610007283817, -2098029341745, 7221561430933, -24875274224531
Offset: 0
a(0) = 1;
a(1) = [x^1] 1/(1 + x) = -1;
a(2) = [x^2] 1/((1 + x)^2*(1 + x^2)) = 2;
a(3) = [x^3] 1/((1 + x)^3*(1 + x^2)^2*(1 + x^3)) = -5;
a(4) = [x^4] 1/((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4)) = 18;
a(5) = [x^5] 1/((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5)) = -60, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + x^k)^(n-k+1) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), 1, -1, 1, -1, ...
n = 2: 1, -2, (2), -2, 3, -4, ...
n = 3: 1, -3, 4, (-5), 9, -14, ...
n = 4: 1, -4, 7, -10, (18), -30, ...
n = 5: 1, -5, 11, -18, 33, (-60), ...
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Table[SeriesCoefficient[Product[1/(1 + x^k)^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 27}]
Showing 1-3 of 3 results.
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