cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A303173 a(n) = [x^n] Product_{k=1..n} (1 - x^k)^(n-k+1).

Original entry on oeis.org

1, -1, 0, 4, -7, 0, 13, 10, -92, 21, 720, -2019, 1193, 6281, -18054, 16111, 11059, -14653, -57685, -86620, 1281406, -3454742, 2383734, 9409968, -30397071, 43327680, -56130326, 128981571, -73487834, -1219918457, 5059678044, -7826243881, -4131571113, 38850603452
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

Examples

			a(0) = 1;
a(1) = [x^1] (1 - x) = -1;
a(2) = [x^2] (1 - x)^2*(1 - x^2) = 0;
a(3) = [x^3] (1 - x)^3*(1 - x^2)^2*(1 - x^3) = 4;
a(4) = [x^4] (1 - x)^4*(1 - x^2)^3*(1 - x^3)^2*(1 - x^4) = -7;
a(5) = [x^5] (1 - x)^5*(1 - x^2)^4*(1 - x^3)^3*(1 - x^4)^2*(1 - x^5) = 0, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} (1 - x^k)^(n-k+1) begins:
n = 0: (1),  0,  0,  0,    0,   0,  ...
n = 1:  1, (-1), 0,  0,    0,   0,  ...
n = 2:  1,  -2, (0), 2,   -1,   0,  ...
n = 3:  1,  -3,  1, (4),  -2,  -2,  ...
n = 4:  1,  -4,  3,  6,  (-7), -2,  ...
n = 5:  1,  -5,  6,  7,  -16,  (0), ...

Links

Crossrefs

Cf. A008705, A073592, A206228, A206229, A303174.

Programs

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

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

Views

Author

Ilya Gutkovskiy, Apr 19 2018

Keywords

Examples

			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), ...

Links

Crossrefs

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

Programs

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

Formula

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

A318204 Decimal expansion of a constant related to the asymptotics of A255526.

Original entry on oeis.org

3, 5, 0, 9, 7, 5, 4, 3, 2, 7, 9, 4, 9, 7, 0, 3, 3, 4, 0, 4, 3, 7, 2, 7, 3, 5, 2, 3, 3, 7, 5, 1, 9, 3, 6, 9, 8, 4, 5, 4, 7, 8, 9, 7, 3, 3, 9, 3, 1, 7, 3, 9, 9, 1, 1, 7, 8, 9, 8, 9, 9, 3, 7, 8, 5, 8, 5, 4, 8, 2, 1, 7, 0, 1, 5, 1, 2, 0, 0, 7, 7, 4, 4, 5, 6, 4, 8, 9, 4, 0, 8, 1, 3, 0, 7, 5, 1, 2, 1, 3, 2, 6, 4, 0, 2
Offset: 1

Views

Author

Vaclav Kotesovec, Aug 21 2018

Keywords

Examples

			3.509754327949703340437273523375193698454789733931739911...

Crossrefs

Cf. A255526, A278428, A303174.

Programs

  • Mathematica
    RealDigits[1/r /. FindRoot[{2*r == s*QPochhammer[-1, -s], 2*r == s^2*Derivative[0, 1][QPochhammer][-1, -s]}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Oct 03 2023 *)

A303483 a(n) = [x^n] Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1).

Original entry on oeis.org

1, 2, 10, 64, 436, 3072, 22096, 161148, 1187118, 8812050, 65806720, 493827256, 3720698056, 28128081912, 213258301824, 1620878656280, 12346263051028, 94221026620572, 720267101230410, 5514346833878672, 42274910234115352, 324490877248800232, 2493471670778297856, 19179885230907692452
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 24 2018

Keywords

Examples

			a(0) = 1;
a(1) = [x^1] (1 + x)/(1 - x) = 2;
a(2) = [x^2] ((1 + x)^2*(1 + x^2))/((1 - x)^2*(1 - x^2)) = 10;
a(3) = [x^3] ((1 + x)^3*(1 + x^2)^2*(1 + x^3))/((1 - x)^3*(1 - x^2)^2*(1 - x^3)) = 64;
a(4) = [x^4] ((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4))/((1 - x)^4*(1 - x^2)^3*(1 - x^3)^2*(1 - x^4)) = 436;
a(5) = [x^5] ((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5))/((1 - x)^5*(1 - x^2)^4*(1 - x^3)^3*(1 - x^4)^2*(1 - x^5)) = 3072, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} ((1 + x^k)/(1 - x^k))^(n-k+1) begins:
n = 0: (1),  0,   0,    0,    0,     0,  ...
n = 1:  1,  (2),  2,    2,    2,     2,  ...
n = 2:  1,   4, (10),  20,   34,    52,  ...
n = 3:  1,   6,  22,  (64), 158,   346,  ...
n = 4:  1,   8,  38,  140, (436), 1200,  ...
n = 5:  1,  10,  58,  256,  946, (3072), ...

Crossrefs

Cf. A206228, A206229, A270919, A303173, A303174.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

Formula

a(n) ~ c * d^n / sqrt(n), where d = 7.862983395705905261519347909953827161057584... and c = 0.23317816342157644853479309078... - Vaclav Kotesovec, May 04 2018
Showing 1-4 of 4 results.

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