A303175
a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).
Original entry on oeis.org
1, 1, 5, 34, 322, 3803, 55297, 953815, 19086057, 434477488, 11086102633, 313318606066, 9714265351819, 327788649292844, 11957321196905337, 468872400449456885, 19666225828334583690, 878560858388253803180, 41645712575272737701666, 2087686693048676581394052
Offset: 0
a(0) = 1;
a(1) = [x^1] 1/(1 - x) = 1;
a(2) = [x^2] 1/((1 - 2*x)*(1 - x^2)) = 5;
a(3) = [x^3] 1/((1 - 3*x)*(1 - 2*x^2)*(1 - x^3)) = 34;
a(4) = [x^4] 1/((1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4)) = 322;
a(5) = [x^5] 1/((1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5)) = 3803, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 - (n - k + 1)*x^k) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (1), 1, 1, 1, 1, ...
n = 2: 1, 2, (5), 10, 21, 42, ...
n = 3: 1, 3, 11, (34), 106, 320, ...
n = 4: 1, 4, 19, 78, (322), 1294, ...
n = 5: 1, 5, 29, 148, 758, (3803), ...
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Table[SeriesCoefficient[Product[1/(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]
A303190
a(n) = [x^n] Product_{k=1..n} 1/(1 + (n - k + 1)*x^k).
Original entry on oeis.org
1, -1, 3, -22, 224, -2759, 41629, -743319, 15285861, -355719616, 9242332881, -265191971970, 8328195163545, -284124989856012, 10463788330880961, -413744821089831397, 17482192791456272614, -786119610413822514764, 37482612103603819839034, -1888918995730788198553380
Offset: 0
a(0) = 1;
a(1) = [x^1] 1/(1 + x) = -1;
a(2) = [x^2] 1/((1 + 2*x)*(1 + x^2)) = 3;
a(3) = [x^3] 1/((1 + 3*x)*(1 + 2*x^2)*(1 + x^3)) = -22;
a(4) = [x^4] 1/((1 + 4*x)*(1 + 3*x^2)*(1 + 2*x^3)*(1 + x^4)) = 224;
a(5) = [x^5] 1/((1 + 5*x)*(1 + 4*x^2)*(1 + 3*x^3)*(1 + 2*x^4)*(1 + x^5)) = -2759, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + (n - k + 1)*x^k) begins:
n = 0: (1), 0, 0, 0, 0, 0, ...
n = 1: 1, (-1), 1, -1, 1, -1, ...
n = 2: 1, -2, (3), -6, 13, -26, ...
n = 3: 1, -3, 7, (-22), 70, -208, ...
n = 4: 1, -4, 13, -54, (224), -890, ...
n = 5: 1, -5, 21, -108, 554, (-2759), ...
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Table[SeriesCoefficient[Product[1/(1 + (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]
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.
...
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}]
Showing 1-3 of 3 results.