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}]
A386720
a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^3) is the g.f. of A023872.
Original entry on oeis.org
1, 1, 19, 163, 1571, 15276, 152029, 1525420, 15460771, 157716235, 1617959044, 16672687769, 172459185341, 1789587777849, 18621317408384, 194222638392213, 2029985619026851, 21256104343844595, 222937740908641405, 2341629730618924374, 24627719497316157396, 259326672761381979574
Offset: 0
-
with(numtheory):
G(x) := series(exp(add(sigma[4](k)*x^k/k, k = 1..25)), x, 26):
seq(coeftayl(G(x)^n, x = 0, n), n = 0..25);
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Table[SeriesCoefficient[Product[1/(1-x^k)^(n*k^3), {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[DivisorSigma[4, k]*x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 25}]
A300457
a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n^k).
Original entry on oeis.org
1, -1, -3, -1, 25, 624, 9871, 170470, 3027249, 55077245, 979330606, 15079702923, 94670678245, -7958168036625, -626145997536240, -34564907982551791, -1733699815491494303, -84294315853736719077, -4067859614343931897505, -196552300464314521511610, -9519733465269825759734169
Offset: 0
The table of coefficients of x^k in expansion of Product_{k>=1} (1 - x^k)^(n^k) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), -1, 0, 0, 1, ...
n = 2: 1, -2, (-3), 0, 2, 12, ...
n = 3: 1, -3, -6, (-1), 9, 63, ...
n = 4: 1, -4, -10, -4, (25), 224, ...
n = 5: 1, -5, -15, -10, 55, (624), ...
Cf.
A010815,
A008705,
A252654,
A252782,
A255672,
A270917,
A270922,
A281266,
A281267,
A281268,
A283333,
A292805,
A300456,
A300458.
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Table[SeriesCoefficient[Product[(1 - x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
A300458
a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n^k).
Original entry on oeis.org
1, -1, -1, -10, 11, 374, 9792, 183847, 3469427, 65038049, 1195396233, 19667738452, 189089161562, -6219720781782, -606316892131934, -35104997710496175, -1795953382595105853, -88223902016631657740, -4283800987347611165184, -207864171877269042498096, -10102590396625592962089500
Offset: 0
The table of coefficients of x^k in expansion of Product_{k>=1} 1/(1 + x^k)^(n^k) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), 0, -1, 1, -1, ...
n = 2: 1, -2, (-1), -4, 3, -2, ...
n = 3: 1, -3, -3, (-10), 6, 15, ...
n = 4: 1, -4, -6, -20, (11), 104, ...
n = 5: 1, -5, -10, -35, 20, (374), ...
Cf.
A081362,
A252654,
A255526,
A252782,
A255672,
A270917,
A270922,
A281266,
A281267,
A281268,
A283333,
A292805,
A300456,
A300457.
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Table[SeriesCoefficient[Product[1/(1 + x^k)^(n^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]