A298987
a(n) = [x^n] Product_{k>=1} (1 + n*x^k)^k.
Original entry on oeis.org
1, 1, 4, 27, 80, 400, 1908, 6223, 31296, 116478, 450100, 1828915, 7360848, 26906828, 95776772, 403908975, 1421758720, 5072014447, 18481180644, 68350964211, 246180936400, 827642046294, 2958748580084, 10294629775620, 36607347335232, 120800714172500, 407951731319860, 1405943613730899
Offset: 0
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Table[SeriesCoefficient[Product[(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 27}]
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.