A299033
a(n) = n! * [x^n] Product_{k>=1} (1 - x^k)^(n/k).
Original entry on oeis.org
1, -1, 0, 15, -136, 885, -4896, 43085, -787200, 7775271, 326355200, -22138191801, 781498160640, -18924340012435, 239123351330304, 5915023788331125, -568462201562300416, 25327272129182225295, -795994018378027868160, 15538852668590468027711
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, 1, -1, 41, -131, ...
n = 2: 1, -2, (0), 8, -4, 72, -704, ...
n = 3: 1, -3, 3, (15), -45, 63, -1539, ...
n = 4: 1, -4, 8, 16, (-136), 224, -1856, ...
n = 5: 1, -5, 15, 5, -265, (885), -2075, ...
n = 6: 1, -6, 24, -24, -396, 2376, (-4896), ...
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Table[n! SeriesCoefficient[Product[(1 - x^k)^(n/k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]
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}]
A319176
a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k/k)^n.
Original entry on oeis.org
1, 1, 8, 93, 1532, 32240, 829284, 25192454, 882825936, 35055329832, 1555548490560, 76285107738312, 4097094075364608, 239167754501235456, 15077741379436233120, 1020918130521930465120, 73892194568147257761024, 5693112248722998479169408, 465208700406183224884224000
Offset: 0
-
Table[n! SeriesCoefficient[Product[1/(1 - x^k/k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[n Sum[Sum[x^(j k)/(k j^k), {j, 1, n}], {k, 1, n}]], {x, 0, n}], {n, 0, 18}]
Showing 1-4 of 4 results.