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

0,3

Links

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

Formula

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

Example

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), ...

Mathematica

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

Crossrefs

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

Sequence in context: A068475 A097817 A344556 * A357422 A197714 A198078

Adjacent sequences: A303172 A303173 A303174 * A303176 A303177 A303178

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 19 2018

Status

approved