A206229 -id:A206229 - cp's OEIS frontend

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

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

Offset: 1

Vaclav Kotesovec, Mar 25 2016

This constant is very close to exp(5*Pi/(6*sqrt(2))) / sqrt(2) = 4.502476748630924546525119125234175537729... - Vaclav Kotesovec, May 17 2018

			4.502476747617354487738593932700784406763128756091621633464540424...

Cf. A181315, A206229, A270913, A270915, A303071, A304626, A327280.

RealDigits[1/r /. FindRoot[{2*s == QPochhammer[-1, r*s], r*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/2}, {s, 1/2}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A270913(n)^(1/n).

A206228 a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^(n-k+1).

Original entry on oeis.org

1, 1, 4, 17, 80, 384, 1887, 9385, 47139, 238488, 1213588, 6204547, 31844710, 163978344, 846741721, 4382945317, 22735196277, 118151632006, 615032941924, 3206257881171, 16736910271178, 87472908459696, 457662760258109, 2396899780970552, 12564645719730297

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Number of partitions of n with 1 kind of n's, 2 kinds of (n-1)'s, ..., n kinds of 1's, see example. [Joerg Arndt, May 17 2013]

			Let [x^n] F(x) denote the coefficient of x^n in F(x); then
a(0) = 1;
a(1) = [x] 1/(1-x) = 1;
a(2) = [x^2] 1/((1-x)^2*(1-x^2)) = 4;
a(3) = [x^3] 1/((1-x)^3*(1-x^2)^2*(1-x^3)) = 17;
a(4) = [x^4] 1/((1-x)^4*(1-x^2)^3*(1-x^3)^2*(1-x^4)) = 80; ...
as illustrated below.
The coefficients in Product_{k=1..n} 1/(1-x^k)^(n-k+1) for n=0..9 begin:
n=0: [(1), 0, 0, 0, 0, 0, 0, ...];
n=1: [1,(1), 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...];
n=2: [1, 2,(4), 6, 9, 12, 16, 20, 25, 30, 36, 42, ...]; (A002620)
n=3: [1, 3, 8, (17), 33, 58, 97, 153, 233, 342, 489, 681, ...]; (A002625)
n=4: [1, 4, 13, 34, (80), 170, 339, 636, 1141, 1964, 3270, ...];
n=5: [1, 5, 19, 58, 157,(384), 874, 1869, 3803, 7408, 13907, ...];
n=6: [1, 6, 26, 90, 273, 746, (1887), 4474, 10062, 21620, ...];
n=7: [1, 7, 34, 131, 438, 1314, 3632, (9385), 22940, 53466, ...];
n=8: [1, 8, 43, 182, 663, 2158, 6445, 17944, (47139), 117842, ...];
n=9: [1, 9, 53, 244, 960, 3361, 10757, 32008, 89651, (238488), ...]; ...
where the coefficients in parenthesis start this sequence.
Incidentally, the antidiagonal sums in the above table form A206119.
From _Joerg Arndt_, May 17 2013: (Start)
There are a(3)=17 partitions of 3 into 1 kind of 3's, 2 kinds of 2's, and 3 kinds of 1's:
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:0  1:2  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:0  1:1  1:2  ]
06:  [ 1:0  1:2  1:2  ]
07:  [ 1:0  2:0  ]
08:  [ 1:0  2:1  ]
09:  [ 1:1  1:1  1:1  ]
10:  [ 1:1  1:1  1:2  ]
11:  [ 1:1  1:2  1:2  ]
12:  [ 1:1  2:0  ]
13:  [ 1:1  2:1  ]
14:  [ 1:2  1:2  1:2  ]
15:  [ 1:2  2:0  ]
16:  [ 1:2  2:1  ]
17:  [ 3:0  ]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A206119, A206229.

Table[SeriesCoefficient[Product[1/(1 - x^k)^(n-k+1), {k, 1, n}], {x, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Aug 21 2018 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^n))^(n-k+1)),n)}
for(n=0,41,print1(a(n),", "))

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724158141653278798514832712869470973196907560641... and c = 0.2030089852709942695768237484498370155967795685257713505678384193773498... - Vaclav Kotesovec, Aug 21 2018

A303174 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).

Original entry on oeis.org

1, -1, 2, -5, 18, -60, 189, -601, 1967, -6544, 21872, -73247, 246080, -829924, 2808357, -9527485, 32389671, -110316862, 376372802, -1286063899, 4400499380, -15075608840, 51704898623, -177513230200, 610007283817, -2098029341745, 7221561430933, -24875274224531

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 + x)^2*(1 + x^2)) = 2;
a(3) = [x^3] 1/((1 + x)^3*(1 + x^2)^2*(1 + x^3)) = -5;
a(4) = [x^4] 1/((1 + x)^4*(1 + x^2)^3*(1 + x^3)^2*(1 + x^4)) = 18;
a(5) = [x^5] 1/((1 + x)^5*(1 + x^2)^4*(1 + x^3)^3*(1 + x^4)^2*(1 + x^5)) = -60, etc.
...
The table of coefficients of x^k in expansion of Product_{k=1..n} 1/(1 + x^k)^(n-k+1) begins:
n = 0: (1),  0,   0,    0,   0,    0,  ...
n = 1:  1, (-1),  1,   -1,   1,   -1,  ...
n = 2:  1,  -2,  (2),  -2,   3,   -4,  ...
n = 3:  1,  -3,   4,  (-5),  9,  -14,  ...
n = 4:  1,  -4,   7,  -10, (18), -30,  ...
n = 5:  1,  -5,  11,  -18,  33, (-60), ...

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

Cf. A206228, A206229, A255526, A255528, A303173.

Table[SeriesCoefficient[Product[1/(1 + x^k)^(n - k + 1), {k, 1, n}], {x, 0, n}], {n, 0, 27}]

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = A318204 = 3.50975432794970334043727352337... and c = 0.2457469629428839220188283... - Vaclav Kotesovec, Aug 21 2018

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

Ilya Gutkovskiy, Apr 19 2018

sign

			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), ...

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

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

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

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}]

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

Ilya Gutkovskiy, Apr 24 2018

nonn

			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), ...

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

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

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

cp's OEIS Frontend

A270914 Decimal expansion of a constant related to the asymptotics of A270913.

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A206228 a(n) = [x^n] Product_{k=1..n} 1/(1 - x^k)^(n-k+1).

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A303174 a(n) = [x^n] Product_{k=1..n} 1/(1 + x^k)^(n-k+1).

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

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

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

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