A298985
a(n) = [x^n] Product_{k>=1} 1/(1 - n*x^k)^k.
Original entry on oeis.org
1, 1, 8, 54, 496, 5400, 73728, 1204322, 23167808, 512093178, 12781430600, 355128859129, 10863077554224, 362572265689777, 13107541496092960, 510105773344747725, 21258690342206888192, 944467894258279964254, 44555341678790400325512, 2224158766859058600584834, 117123916650423288611260400
Offset: 0
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b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(i+j-1, j)*b(n-i*j, i-1, k)*k^j, j=0..n/i)))
end:
a:= n-> b(n$3):
seq(a(n), n=0..30); # Alois P. Heinz, Sep 23 2018
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Table[SeriesCoefficient[Product[1/(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
A298988
a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.
Original entry on oeis.org
1, -1, 0, -18, 208, -2400, 36504, -663754, 13808320, -324176418, 8487126400, -245122390601, 7741417124880, -265402847130421, 9816338228638872, -389618889514254225, 16518399076342421248, -745025763154442071130, 35619835529954597786208, -1799459812004380374518790, 95780758238408017088795600
Offset: 0
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Table[SeriesCoefficient[Product[1/(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]
A298986
a(n) = [x^n] Product_{k>=1} (1 - n*x^k)^k.
Original entry on oeis.org
1, -1, -4, 9, 48, 100, -756, -3479, -1600, 24462, 225900, 364573, -643536, -9251736, -36989316, -32397975, 165039872, 1725828525, 5338814652, 8082713829, -26321848400, -233434232766, -811526778964, -1731126953532, 1151302859712, 23632432765000, 113461901639788, 287935019845749
Offset: 0
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Table[SeriesCoefficient[Product[(1 - n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}]
A300412
a(n) = [x^n] Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k.
Original entry on oeis.org
1, 2, 16, 144, 1376, 15800, 210816, 3333372, 61688448, 1318588146, 32004369200, 869282342632, 26099925704928, 857736429098848, 30605729417479104, 1177841009504482200, 48614265201514729984, 2141639401723095243324, 100282931820560447963568, 4973060138191518242569120
Offset: 0
The table of coefficients of x^k in expansion of Product_{k>=1} ((1 + n*x^k)/(1 - n*x^k))^k begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (2), 6, 16, 38, 88, ...
n = 2: 1, 4, (16), 60, 192, 596, ...
n = 3: 1, 6, 30, (144), 582, 2280, ...
n = 4: 1, 8, 48, 280, (1376), 6568, ...
n = 5: 1, 10, 70, 480, 2790, (15800), ...
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Table[SeriesCoefficient[Product[((1 + n x^k)/(1 - n x^k))^k, {k, 1, n}], {x, 0, n}], {n, 0, 19}]
Showing 1-4 of 4 results.