A022661 -id:A022661 - cp's OEIS frontend

A336977 Expansion of Product_{k>=1} (1 - x^k * (k + x)).

1, -1, -3, -2, 1, 9, 11, 15, 6, -18, -46, -54, -115, -101, 32, 82, 182, 455, 804, 915, 434, -114, 196, -974, -3507, -6913, -7555, -5081, -4480, -7463, -4861, 7677, 25779, 56080, 76015, 51021, 53996, 104670, 114132, 93920, -33843, -233380, -491936, -658774, -597698, -601576

Offset: 0

Seiichi Manyama, Aug 09 2020

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Seiichi Manyama, Table of n, a(n) for n = 0..1000

Convolution inverse of A336975.

Cf. A022661, A306565, A336976, A336978, A336979, A336980.

m = 45; CoefficientList[Series[Product[1 - x^k*(k + x), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 01 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k*(k+x)))

N=66; x='x+O('x^N); Vec(exp(-sum(k=1, N, x^k*sumdiv(k, d, (k/d+x)^d/d))))

G.f.: exp( - Sum_{k>=1} x^k * Sum_{d|k} (k/d + x)^d / d).

A292164 Expansion of Product_{k>=1} (1 - k^2*x^k).

Original entry on oeis.org

1, -1, -4, -5, -7, 27, 17, 167, 110, -42, 10, -706, -4001, -3915, 3079, -18640, 9869, 21403, 130565, 107250, -15661, 420664, 599540, -161785, -1232833, -5836888, -5129796, 6516714, -29068180, -14953045, -41490510, 20261320, 30395771, 441235155, 205289550

Offset: 0

Seiichi Manyama, Sep 10 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Column k=2 of A292166.

Cf. A022629, A022661, A077335, A092484.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i$2)*a(i$2), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
    b[n, i - 1] + If[i > n, 0, i^2*b[n - i, i]]]];
a[n_] := a[n] = If[n == 0, 1,
    -Sum[b[n - i, n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 04 2024, after Alois P. Heinz *)

N=66; x='x+O('x^N); Vec(prod(n=1, N, 1-n^2*x^n))

Convolution inverse of A077335.

G.f.: exp(-Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-jx^j)^(j^(kj)) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -8, -1, 1, -1, -32, -73, -1, 1, -1, -128, -2155, -919, 5, 1, -1, -512, -58921, -259477, -13977, 1, 1, -1, -2048, -1593811, -67041751, -48496477, -253640, 13, 1, -1, -8192, -43044673, -17178144301, -152513231553, -13001163543, -5290184, 4

Offset: 0

Seiichi Manyama, Nov 04 2017

			Square array begins:
    1,    1,       1,         1,            1, ...
   -1,   -1,      -1,        -1,           -1, ...
   -2,   -8,     -32,      -128,         -512, ...
   -1,  -73,   -2155,    -58921,     -1593811, ...
   -1, -919, -259477, -67041751, -17178144301, ...

Seiichi Manyama, Antidiagonals n = 0..52, flattened

Columns k=0..2 give A022661, A294606, A294607.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A283675, A294609.

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k*d+1+j/d)) * A(n-j,k) for n > 0.

A294587 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^(j^k).

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -4, -1, 1, -1, -8, -5, -1, 1, -1, -16, -19, -3, 5, 1, -1, -32, -65, -13, 23, 1, 1, -1, -64, -211, -63, 131, 44, 13, 1, -1, -128, -665, -301, 815, 497, 104, 4, 1, -1, -256, -2059, -1383, 5195, 4840, 1149, 70, 0, 1, -1, -512, -6305, -6133, 33143, 45021, 13752, 662, -93, 2

Offset: 0

Seiichi Manyama, Nov 03 2017

			Square array begins:
    1,  1,   1,   1,    1, ...
   -1, -1,  -1,  -1,   -1, ...
   -2, -4,  -8, -16,  -32, ...
   -1, -5, -19, -65, -211, ...
   -1, -3, -13, -63, -301, ...

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

Columns k=0..2 give A022661, A266964, A294588.
Rows n=0..1 give A000012, (-1)*A000012.
Cf. A283272.

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(k+1+j/d)) * A(n-j,k) for n > 0.

A303189 a(n) = [x^n] Product_{k=1..n} (1 - (n - k + 1)*x^k).

Original entry on oeis.org

1, -1, -1, 5, 7, 21, -94, -117, -404, -840, 3541, 4536, 14412, 31313, 72175, -249424, -262828, -930639, -1895460, -4441316, -8085972, 24112570, 26214408, 87131883, 180197979, 411759028, 748154122, 1525043990, -3554837744, -3210408245, -11955482059, -23817949142, -55221348072

Offset: 0

Ilya Gutkovskiy, Apr 19 2018

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			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) = 5;
a(4) = [x^4] (1 - 4*x)*(1 - 3*x^2)*(1 - 2*x^3)*(1 - x^4) = 7;
a(5) = [x^5] (1 - 5*x)*(1 - 4*x^2)*(1 - 3*x^3)*(1 - 2*x^4)*(1 - x^5) = 21, 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,  (5),  3,   2,  ...
n = 4:  1,  -4,  -3,  10,  (7), 10,  ...
n = 5:  1,  -5,  -4,  17,  13, (21), ...

Cf. A022661, A292132, A303173, A303175, A303188, A303190.

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

A292317 Expansion of Product_{k>=1} ((1 - kx^k)/(1 + kx^k)).

Original entry on oeis.org

1, -2, -2, 0, 6, 8, 0, 4, -10, -50, -36, 16, 12, 80, 44, 88, 390, 180, -94, -712, -1624, -312, -688, 1476, 4444, -6954, -5812, 3816, 7728, 36600, 25708, -13308, -53586, -127048, 10104, 120936, 73490, 157400, -395168, -529472, 833888, 265916, 19300, -1132576

Offset: 0

Seiichi Manyama, Sep 14 2017

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

Cf. A022661, A022693, A265758.

nmax = 50; CoefficientList[Series[Product[(1 - k*x^k)/(1 + k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2018 *)

Convolution of A022661 and A022693.

Convolution inverse of A265758.

A269339 Expansion of Product_{k>=1} (1 - k*x^k) / (1 + x^k).

Original entry on oeis.org

1, -2, -1, 0, 2, 6, -3, 12, -13, -2, -15, 2, -65, 44, 37, -90, 134, 26, 334, -270, 66, 18, 774, -1280, -15, -2266, 2627, -352, -3575, -516, -484, 5660, -3629, 21408, -20639, -1228, 15595, 31796, -22214, 55390, -104447, 58958, -160254, 180704, 17402, -103200

Offset: 0

Vaclav Kotesovec, Feb 24 2016

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Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A022661, A081362, A267004, A267005, A268500.

nmax=50; CoefficientList[Series[Product[(1-k*x^k)/(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

cp's OEIS Frontend

A336977 Expansion of Product_{k>=1} (1 - x^k * (k + x)).

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A292164 Expansion of Product_{k>=1} (1 - k^2*x^k).

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A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j*x^j)^(j^(k*j)) in powers of x.

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A294587 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j*x^j)^(j^k).

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A303189 a(n) = [x^n] Product_{k=1..n} (1 - (n - k + 1)*x^k).

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A292317 Expansion of Product_{k>=1} ((1 - k*x^k)/(1 + k*x^k)).

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Mathematica

Formula

A269339 Expansion of Product_{k>=1} (1 - k*x^k) / (1 + x^k).

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Mathematica

A294605 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-jx^j)^(j^(kj)) in powers of x.

A292317 Expansion of Product_{k>=1} ((1 - kx^k)/(1 + kx^k)).