A299034
a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k)^(n/k).
Original entry on oeis.org
1, 1, 8, 93, 1544, 32615, 843264, 25739539, 906373376, 36163950849, 1612483625600, 79458277381901, 4288069172500992, 251520785449249927, 15932801526165085184, 1084003570689331039875, 78835487923639854792704, 6103175938145968656408641, 501114006272655771562911744
Offset: 0
The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(n/k) begins:
n = 0: (1), 0, 0, 0, 0, 0, 0, ...
n = 1: 1, (1), 3, 11, 59, 339, 2629, ...
n = 2: 1, 2, (8), 40, 260, 1928, 17056, ...
n = 3: 1, 3, 15, (93), 711, 6237, 62901, ...
n = 4: 1, 4, 24, 176, (1544), 15456, 174784, ...
n = 5: 1, 5, 35, 295, 2915, (32615), 407725, ...
n = 6: 1, 6, 48, 456, 5004, 61704, (843264), ...
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Table[n! SeriesCoefficient[Product[1/(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]
A300187
a(n) = n! * [x^n] Product_{k>=1} (1 + x^k)^(n/k).
Original entry on oeis.org
1, 1, 4, 39, 488, 7615, 147024, 3371137, 89079808, 2665537713, 89142430400, 3295096700071, 133399600068096, 5870116973678191, 278971698167158528, 14239859507270510625, 776985219329347518464, 45130494178637796970273, 2780224621391401396134912, 181059775626543107582734183
Offset: 0
The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} (1 + x^k)^(n/k) begins:
n = 0: (1), 0, 0, 0, 0, 0, 0, ...
n = 1: 1, (1), 1, 5, 11, 59, 439, ...
n = 2: 1, 2, (4), 16, 68, 328, 2416, ...
n = 3: 1, 3, 9, (39), 207, 1197, 8811, ...
n = 4: 1, 4, 16, 80, (488), 3296, 25984, ...
n = 5: 1, 5, 25, 145, 995, (7615), 65575, ...
n = 6: 1, 6, 36, 240, 1836, 15624, (147024), ...
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Table[n! SeriesCoefficient[Product[(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]
A300188
a(n) = n! * [x^n] Product_{k>=1} 1/(1 + x^k)^(n/k).
Original entry on oeis.org
1, -1, 4, -39, 536, -9115, 185904, -4461877, 123647488, -3886461081, 136538590400, -5300491027711, 225313697972736, -10409021924850211, 519298241645107456, -27824560148201248125, 1593597443825288904704, -97153909607626767338353, 6281720886474120790582272
Offset: 0
The table of coefficients of x^k in expansion of e.g.f. Product_{k>=1} 1/(1 + x^k)^(n/k) begins:
n = 0: (1), 0, 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), 1, -5, 23, -119, 619, ...
n = 2: 1, -2, (4), -16, 92, -568, 3856, ...
n = 3: 1, -3, 9, (-39), 243, -1737, 13671, ...
n = 4: 1, -4, 16, -80, (536), -4256, 37504, ...
n = 5: 1, -5, 25, -145, 1055, (-9115), 88075, ...
n = 6: 1, -6, 36, -240, 1908, -17784, (185904), ...
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Table[n! SeriesCoefficient[Product[1/(1 + x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Showing 1-3 of 3 results.