A286957 -id:A286957 - cp's OEIS frontend

A291698 a(n) = [x^n] Product_{k>=1} (1 + n*x^k).

1, 1, 2, 12, 20, 55, 294, 497, 1224, 2520, 14410, 21912, 54300, 104286, 220710, 1105215, 1697552, 3839382, 7356762, 14873580, 26275620, 132112596, 188666126, 423247104, 772560600, 1535398150, 2632049290, 4975242048, 21273166572, 30649985160, 64824339630, 116604788800, 223181224992

Offset: 0

Ilya Gutkovskiy, Aug 30 2017

The number of partitions of n into distinct parts where each part can be colored in n different ways. For example, there are 4 partitions of 6 into distinct parts, namely 6, 5 + 1, 4 + 2 and 3 + 2 + 1; allowing for the colorings gives a(6) = 6 + 6*6 + 6*6 + 6*6*6 = 294. - Peter Bala, Aug 31 2017

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)

Main diagonal of A286957.

Cf. A022629, A124577, A292190, A292304.

seq(coeff(mul(1+n*x^k,k=1..n),x,n),n=0..50); # Robert Israel, Aug 30 2017

Table[SeriesCoefficient[Product[1 + n x^k, {k, 1, n}], {x, 0, n}], {n, 0, 32}]
Table[SeriesCoefficient[QPochhammer[-n, x]/(1 + n), {x, 0, n}], {n, 0, 32}]

a(n) = A286957(n,n).
a(n) == 0 (mod n); a(n) == n (mod n^2). - Peter Bala, Aug 31 2017
Conjecture: a(n) ~ exp(sqrt(2*(log(n)^2 + Pi^2/3)*n)) * (log(n)^2 + Pi^2/3)^(1/4) / (sqrt(Pi) * (2*n)^(5/4)). - Vaclav Kotesovec, Sep 15 2017

A292131 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - k*x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -2, 0, 0, 1, -4, -3, 2, 0, 0, 1, -5, -4, 6, 2, 1, 0, 1, -6, -5, 12, 6, 6, 0, 0, 1, -7, -6, 20, 12, 15, -2, 1, 0, 1, -8, -7, 30, 20, 28, -12, 2, 0, 0, 1, -9, -8, 42, 30, 45, -36, -3, -6, 0, 0, 1, -10, -9, 56, 42, 66, -80

Offset: 0

Seiichi Manyama, Sep 09 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -2, -3, -4, ...
   0, -1, -2, -3, -4, ...
   0,  0,  2,  6, 12, ...
   0,  0,  2,  6, 12, ...

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

Columns k=0..3 give A000007, A010815, A070877, A292128.
Rows n=0..1 give A000012, (-1)*A001477.
Main diagonal gives A292132.
Cf. A246935, A286957.

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.

A382993 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} phi(n/d) * (-k)^d.

Original entry on oeis.org

1, 2, 0, 3, -1, 1, 4, -3, 4, 0, 5, -6, 11, -4, 1, 6, -10, 24, -21, 8, 0, 7, -15, 45, -66, 51, -10, 1, 8, -21, 76, -160, 208, -119, 20, 0, 9, -28, 119, -330, 629, -676, 315, -34, 1, 10, -36, 176, -609, 1560, -2590, 2344, -831, 60, 0, 11, -45, 249, -1036, 3367, -7750, 11165, -8226, 2195, -100, 1

Offset: 1

Seiichi Manyama, Apr 11 2025

			Square array begins:
  1,   2,    3,    4,     5,     6,      7, ...
  0,  -1,   -3,   -6,   -10,   -15,    -21, ...
  1,   4,   11,   24,    45,    76,    119, ...
  0,  -4,  -21,  -66,  -160,  -330,   -609, ...
  1,   8,   51,  208,   629,  1560,   3367, ...
  0, -10, -119, -676, -2590, -7750, -19565, ...
  1,  20,  315, 2344, 11165, 39996, 117655, ...

Columns k=1..5 give A000035, (-1)^(n+1) * A074763(n), A343465, A343466, A343467.
Main diagonal gives A382998.
Cf. A000010, A075195, A286957, A382994, A383011.

a(n, k) = -sumdiv(n, d, eulerphi(n/d)*(-k)^d)/n;

A(n,k) = (1/n) * A382994(n,k).
A(n,k) = -(1/n) * Sum_{j=1..n} (-k)^gcd(n,j).
G.f. of column k: Sum_{j>=1} phi(j) * log(1 + k*x^j) / j.
Product_{n>=1} 1/(1 - x^n)^A(n,k) = Product_{n>=1} (1 + k*x^n).

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).

Original entry on oeis.org

1, 2, 5, 19, 49, 126, 469, 1177, 2881, 6481, 23101, 53725, 127153, 274288, 581925, 1860751, 4155649, 9279791, 19409221, 39839239, 77052401, 229393207, 481747949, 1035561408, 2082441025, 4153434376, 7822058869, 14686515649, 39394280689, 79657493191, 163600884901

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

Index entries for sequences related to partitions

Cf. A286957, A291698, A303070, A303071, A303914.

Table[SeriesCoefficient[1/(1 - x) Product[(1 + n x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/(1 - x) Exp[Sum[(-1)^(k + 1) n^k x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[QPochhammer[-n, x]/((1 + n) (1 - x)), {x, 0, n}], {n, 0, 30}]

a(n) = [x^n] (1/(1 - x))*exp(Sum_{k>=1} (-1)^(k+1)*n^k*x^k/(k*(1 - x^k))).

a(n) = Sum_{j=0..n} A286957(j,n).

cp's OEIS Frontend

A291698 a(n) = [x^n] Product_{k>=1} (1 + n*x^k).

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A292131 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - k*x^j).

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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).

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A382993 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} phi(n/d) * (-k)^d.

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A304782 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + n*x^k).

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Programs

Mathematica

Formula

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).