A303190 -id:A303190 - cp's OEIS frontend

A303175 a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).

1, 1, 5, 34, 322, 3803, 55297, 953815, 19086057, 434477488, 11086102633, 313318606066, 9714265351819, 327788649292844, 11957321196905337, 468872400449456885, 19666225828334583690, 878560858388253803180, 41645712575272737701666, 2087686693048676581394052

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

			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), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..384

Cf. A006906, A124577, A206228, A303188, A303189, A303190.

Table[SeriesCoefficient[Product[1/(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 19}]

a(n) ~ n^n * (1 + 1/n + 1/n^2 - 1/n^3 - 3/n^4 - 8/n^5 - 7/n^6 - 13/n^7 + 2/n^8 - 3/n^9 + 31/n^10 + 21/n^11 + 81/n^12 + 2/n^13 + 152/n^14 - 114/n^15 + 173/n^16 - 341/n^17 + 260/n^18 - 936/n^19 + 861/n^20 - 2187/n^21 + 2630/n^22 - 4551/n^23 + 6211/n^24 - 8866/n^25 + 14889/n^26 - 22374/n^27 + 38490/n^28 - 55911/n^29 + 87688/n^30 - ...). - Vaclav Kotesovec, Aug 21 2018

A303188 a(n) = [x^n] Product_{k=1..n} (1 + (n - k + 1)*x^k).

Original entry on oeis.org

1, 1, 1, 7, 9, 23, 148, 221, 526, 1040, 6767, 9664, 23456, 43943, 91363, 499028, 736410, 1650395, 3107540, 6210372, 10819270, 57864166, 80663444, 179915133, 324882691, 640398244, 1087149284, 2039724322, 9121580902, 12913282685, 27250167385, 48645989650, 92634730208, 156124357449

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

nonn

			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) = 7;
a(4) = [x^4] (1 + 4*x)*(1 + 3*x^2)*(1 + 2*x^3)*(1 + x^4) = 9;
a(5) = [x^5] (1 + 5*x)*(1 + 4*x^2)*(1 + 3*x^3)*(1 + 2*x^4)*(1 + x^5) = 23, 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,  (7),  3,   2,  ...
n = 4:  1,  4,  3,  14,  (9), 10,  ...
n = 5:  1,  5,  4,  23,  17, (23), ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A022629, A206229, A291698, A303175, A303189, A303190.

Table[SeriesCoefficient[Product[(1 + (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 33}]

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

Ilya Gutkovskiy, Apr 19 2018

sign

			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), ...

Cf. A022661, A292132, A303173, A303175, A303188, A303190.

Table[SeriesCoefficient[Product[(1 - (n - k + 1) x^k), {k, 1, n}], {x, 0, n}], {n, 0, 32}]

cp's OEIS Frontend

A303175 a(n) = [x^n] Product_{k=1..n} 1/(1 - (n - k + 1)*x^k).

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A303188 a(n) = [x^n] Product_{k=1..n} (1 + (n - k + 1)*x^k).

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Mathematica

A303189 a(n) = [x^n] Product_{k=1..n} (1 - (n - k + 1)*x^k).

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica