A303174
a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).
Original entry on oeis.org
1, -1, 2, -5, 18, -60, 189, -601, 1967, -6544, 21872, -73247, 246080, -829924, 2808357, -9527485, 32389671, -110316862, 376372802, -1286063899, 4400499380, -15075608840, 51704898623, -177513230200, 610007283817, -2098029341745, 7221561430933, -24875274224531
Offset: 0
a(0) = 1;
a(1) = [x^1] 1/(1 + x) = -1;
a(2) = [x^2] 1/((1 + x)^2*(1 + x^2)) = 2;
a(3) = [x^3] 1/((1 + x)^3*(1 + x^2)^2*(1 + x^3)) = -5;
a(4) = [x^4] 1/((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4)) = 18;
a(5) = [x^5] 1/((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5)) = -60, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + x^k)^(n-k+1) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), 1, -1, 1, -1, ...
n = 2: 1, -2, (2), -2, 3, -4, ...
n = 3: 1, -3, 4, (-5), 9, -14, ...
n = 4: 1, -4, 7, -10, (18), -30, ...
n = 5: 1, -5, 11, -18, 33, (-60), ...
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Table[SeriesCoefficient[Product[1/(1 + x^k)^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 27}]
A303189
a(n) = [x^n] Product_{k=1..n} (1 - (n - k + 1)*x^k).
Original entry on oeis.org
1, -1, -1, 5, 7, 21, -94, -117, -404, -840, 3541, 4536, 14412, 31313, 72175, -249424, -262828, -930639, -1895460, -4441316, -8085972, 24112570, 26214408, 87131883, 180197979, 411759028, 748154122, 1525043990, -3554837744, -3210408245, -11955482059, -23817949142, -55221348072
Offset: 0
a(0) = 1;
a(1) = [x^1] (1 - x) = -1;
a(2) = [x^2] (1 - 2*x)*(1 - x^2) = -1;
a(3) = [x^3] (1 - 3*x)*(1 - 2*x^2)*(1 - x^3) = 5;
a(4) = [x^4] (1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4) = 7;
a(5) = [x^5] (1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5) = 21, etc.
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The table of coefficients of x^k in expansion of Product_{k=1..n} (1 - (n - k + 1)*x^k) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), 0, 0, 0, 0, ...
n = 2: 1, -2, (-1), 2, 0, 0 ...
n = 3: 1, -3, -2, (5), 3, 2, ...
n = 4: 1, -4, -3, 10, (7), 10, ...
n = 5: 1, -5, -4, 17, 13, (21), ...
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Table[SeriesCoefficient[Product[(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 32}]
A303483
a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1).
Original entry on oeis.org
1, 2, 10, 64, 436, 3072, 22096, 161148, 1187118, 8812050, 65806720, 493827256, 3720698056, 28128081912, 213258301824, 1620878656280, 12346263051028, 94221026620572, 720267101230410, 5514346833878672, 42274910234115352, 324490877248800232, 2493471670778297856, 19179885230907692452
Offset: 0
a(0) = 1;
a(1) = [x^1] (1 + x)/(1 - x) = 2;
a(2) = [x^2] ((1 + x)^2*(1 + x^2))/((1 - x)^2*(1 - x^2)) = 10;
a(3) = [x^3] ((1 + x)^3*(1 + x^2)^2*(1 + x^3))/((1 - x)^3*(1 - x^2)^2*(1 - x^3)) = 64;
a(4) = [x^4] ((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4))/((1 - x)^4*(1 - x^2)^3*(1 - x^3)^2*(1 - x^4)) = 436;
a(5) = [x^5] ((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5))/((1 - x)^5*(1 - x^2)^4*(1 - x^3)^3*(1 - x^4)^2*(1 - x^5)) = 3072, etc.
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The table of coefficients of x^k in expansion of Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (2), 2, 2, 2, 2, ...
n = 2: 1, 4, (10), 20, 34, 52, ...
n = 3: 1, 6, 22, (64), 158, 346, ...
n = 4: 1, 8, 38, 140, (436), 1200, ...
n = 5: 1, 10, 58, 256, 946, (3072), ...
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Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 23}]
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