A292134 -id:A292134 - cp's OEIS frontend

A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

1, -1, 0, -18, 208, -2400, 36504, -663754, 13808320, -324176418, 8487126400, -245122390601, 7741417124880, -265402847130421, 9816338228638872, -389618889514254225, 16518399076342421248, -745025763154442071130, 35619835529954597786208, -1799459812004380374518790, 95780758238408017088795600

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

sign

Cf. A255528, A261566, A261567, A266971, A292134, A297326, A298985, A298986, A298987.

Table[SeriesCoefficient[Product[1/(1 + n x^k)^k, {k, 1, n}], {x, 0, n}], {n, 0, 20}]

a(n) ~ (-1)^n * n^n * (1 - 2/n + 6/n^2 - 14/n^3 + 33/n^4 - 70/n^5 + 149/n^6 - 298/n^7 + 591/n^8 - 1132/n^9 + 2139/n^10 + ...), for coefficients, see A005380. - Vaclav Kotesovec, Aug 21 2018

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

Ilya Gutkovskiy, Apr 19 2018

sign

			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), ...

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

Cf. A022693, A292134, A303174, A303175, A303188, A303189.

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

a(n) ~ (-1)^n * n^n * (1 - 1/n + 3/n^2 - 7/n^3 + 15/n^4 - 32/n^5 + 65/n^6 - 131/n^7 + 260/n^8 - 501/n^9 + 965/n^10 - 1825/n^11 + 3419/n^12 - 6326/n^13 + 11652/n^14 - 21230/n^15 + 38405/n^16 - 69015/n^17 + 123334/n^18 - 218980/n^19 + 386809/n^20 - 679757/n^21 + 1189360/n^22 - 2071761/n^23 + 3594325/n^24 - 6211826/n^25 + 10698409/n^26 - 18363038/n^27 + 31420994/n^28 - 53605525/n^29 + 91198970/n^30 - ...). - Vaclav Kotesovec, Aug 22 2018

A292133 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 2, -1, 0, 1, -4, 6, -6, 1, 0, 1, -5, 12, -21, 14, -1, 0, 1, -6, 20, -52, 69, -26, 1, 0, 1, -7, 30, -105, 220, -201, 50, -1, 0, 1, -8, 42, -186, 545, -868, 591, -102, 2, 0, 1, -9, 56, -301, 1146, -2705, 3436, -1785, 214, -2, 0

Offset: 0

Seiichi Manyama, Sep 09 2017

			Square array begins:
   1,  1,  1,   1,   1, ...
   0, -1, -2,  -3,  -4, ...
   0,  0,  2,   6,  12, ...
   0, -1, -6, -21, -52, ...
   0,  1, 14,  69, 220, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..3 give A000007, A081362, A071109, A261582.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292134.
Cf. A246935, A286957.

cp's OEIS Frontend

A298988 a(n) = [x^n] Product_{k>=1} 1/(1 + n*x^k)^k.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A303190 a(n) = [x^n] Product_{k=1..n} 1/(1 + (n - k + 1)*x^k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A292133 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + k*x^j).

Views

Author

Keywords

Examples

Links

Crossrefs