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

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

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

• nonn

Examples

			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), ...
		

Links

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

Crossrefs

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

Programs

Mathematica

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