A073592 -id:A073592 - cp's OEIS frontend

A291696 Expansion of Product_{k>=1} (1-x^(k^2))^(k^2).

1, -1, 0, 0, -4, 4, 0, 0, 6, -15, 9, 0, -4, 40, -36, 0, -15, -39, 90, -36, 64, -28, -180, 144, -96, 206, 106, -300, 148, -540, 332, 480, -232, 610, -1029, -189, 114, -86, 1880, -1068, 24, -921, -1545, 2466, -300, 2858, -1514, -3180, 976, -4121, 5590, 1995

Offset: 0

Vaclav Kotesovec, Aug 30 2017

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Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A073592, A291649, A291655.

nmax = 100; CoefficientList[Series[Product[(1-x^(k^2))^(k^2), {k,1,nmax}], {x,0,nmax}], x]

A294580 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j)^j.

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -4, -1, 1, -1, -8, -5, 0, 1, -1, -16, -19, -3, 4, 1, -1, -32, -65, -21, 23, 4, 1, -1, -64, -211, -111, 139, 44, 7, 1, -1, -128, -665, -525, 863, 448, 104, 3, 1, -1, -256, -2059, -2343, 5419, 4316, 1414, 70, -2, 1, -1, -512, -6305, -10101, 34103, 40024, 18164, 1206, -93, -9

Offset: 0

Seiichi Manyama, Nov 02 2017

			Square array begins:
    1,  1,   1,    1,    1, ...
   -1, -1,  -1,   -1,   -1, ...
   -2, -4,  -8,  -16,  -32, ...
   -1, -5, -19,  -65, -211, ...
    0, -3, -21, -111, -525, ...

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

Columns k=0..2 give A073592, A266964, A294581.
Rows n=0..3 give A000012, (-1)*A000012, (-1)*A000079(n+1), (-1)*A001047(n+1).
Cf. A292166, A294582.

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j/d)) * A(n-j,k) for n > 0.

A298986 a(n) = [x^n] Product_{k>=1} (1 - n*x^k)^k.

Original entry on oeis.org

1, -1, -4, 9, 48, 100, -756, -3479, -1600, 24462, 225900, 364573, -643536, -9251736, -36989316, -32397975, 165039872, 1725828525, 5338814652, 8082713829, -26321848400, -233434232766, -811526778964, -1731126953532, 1151302859712, 23632432765000, 113461901639788, 287935019845749

Offset: 0

Ilya Gutkovskiy, Jan 31 2018

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Cf. A073592, A266964, A292132, A297324, A298985, A298987, A298988.

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

A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -5, 2, 15, 12, -36, -92, -17, 167, 358, 283, -293, -1321, -2012, -1101, 2299, 7296, 10505, 6901, -7705, -31240, -52490, -51336, -6032, 91521, 217064, 303776, 250595, -36282, -575622, -1234465, -1684515, -1448538, -66980, 2610835, 6087681, 8990575

Offset: 0

Seiichi Manyama, Mar 03 2017

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

Cf. A283224 (exp( Sum_{n>=1} sigma_2(2*n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(2*n)*x^n/n ): A115110 (k=1), this sequence (k=2).
Cf. exp( Sum_{n>=1} -sigma_2(m*n)*x^n/n ): A073592 (m=1), this sequence (m=2), A283243 (m=3).

a(n) = -(1/n)*Sum_{k=1..n} sigma_2(2*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A283243 Expansion of exp( Sum_{n>=1} -sigma_2(3n)x^n/n ) in powers of x.

Original entry on oeis.org

1, -10, 25, 53, -270, -77, 1057, 610, -2031, -5438, -1953, 17236, 34121, 3351, -103369, -195850, -55471, 468448, 1067785, 764094, -1430780, -4974559, -6242563, 334620, 16946199, 34459888, 29243953, -24503978, -124514921, -205795663, -140256312, 191109263

Offset: 0

Seiichi Manyama, Mar 03 2017

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

Cf. A283237 (sigma_2(3*n)), A283238 (exp( Sum_{n>=1} sigma_2(3*n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(3*n)*x^n/n ): A185654 (k=1), this sequence (k=2).
Cf. exp( Sum_{n>=1} -sigma_2(m*n)*x^n/n ): A073592 (m=1), A283242 (m=2), this sequence (m=3).

A283243_vec(m)=Vec(exp(sum(n=1,m,-sigma(3*n,2)*x^n/n)+x*O(x^m))) \\ Yields m+1 terms a(0..m). - M. F. Hasler, Mar 05 2017

a(n) = -(1/n)*Sum_{k=1..n} sigma_2(3*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A305205 a(n) = [x^n] exp(-Sum_{k>=1} x^k/(k*(1 - x^k)^n)).

Original entry on oeis.org

1, -1, -2, -3, -4, 30, 274, 1841, 9358, 32463, -41557, -2265846, -28939286, -272101778, -2038274408, -10494221259, 9056975574, 1244820826687, 22703501504125, 299864024917632, 3221417281127823, 26849622543478562, 110101743392268978, -1810492304600468063

Offset: 0

Ilya Gutkovskiy, May 27 2018

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N. J. A. Sloane, Transforms

Cf. A073592, A292386, A292387, A293554, A305206.

Table[SeriesCoefficient[Exp[-Sum[x^k/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[(1 - x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} (1 - x^k)^binomial(n+k-2,n-1).

A299019 Expansion of Product_{k>=1} (1 - x^k)^(k+1).

Original entry on oeis.org

1, -2, -2, 2, 3, 6, -1, -2, -10, -14, -7, -2, 11, 26, 43, 30, 28, -6, -40, -92, -128, -132, -115, -48, 54, 200, 339, 484, 499, 476, 274, -32, -501, -998, -1539, -1924, -2042, -1838, -1139, 12, 1664, 3540, 5588, 7258, 8392, 8230, 6812, 3480, -1472, -8150, -15737, -23670, -30478

Offset: 0

Ilya Gutkovskiy, Aug 11 2018

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Convolution of A010815 and A073592.

Convolution inverse of A005380.

Cf. A005380, A010815, A073592, A092346.

nmax = 52; CoefficientList[Series[Product[(1 - x^k)^(k + 1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 52; CoefficientList[Series[Exp[-Sum[(DivisorSigma[1, k] + DivisorSigma[2, k]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[d (d + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

G.f.: exp(-Sum_{k>=1} (sigma_1(k) + sigma_2(k))*x^k/k).

A258412 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^k dx.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 29 2015

Integral_{x=0..1} Product_{k=1..n} (1+x^k)^k dx ~ 3*2^(n*(n+1)/2 + 1)/n^3.
Integral_{x=0..1} Product_{k=1..n} (1+x^k)   dx ~ 2^(n+2)/n^2.
Integral_{x=0..1} Product_{k>=1}   (1-x^k)   dx = A258232 = 0.3684125359314...
Integral_{x=0..1} Product_{k=1..n} (1-x^k)^n dx ~ 1/n.
Integral_{x=0..1} Product_{k=1..n} (1+x^k)^n dx ~ 2^(n^2 + 2)/n^3.

			0.298783365106567298770953772114...

Vaclav Kotesovec, The integration of q-series

Cf. A073592, A258232.

More digits from Vaclav Kotesovec, Oct 10 2023

A298599 Expansion of Product_{k>=2} (1 - x^k)^k.

Original entry on oeis.org

1, 0, -2, -3, -3, 1, 5, 12, 15, 13, 4, -13, -38, -62, -75, -76, -44, 17, 114, 225, 337, 411, 419, 311, 68, -324, -836, -1405, -1947, -2305, -2338, -1865, -787, 1001, 3396, 6261, 9216, 11785, 13281, 13036, 10285, 4502, -4619, -16918, -31657, -47463, -62182, -73112, -76925

Offset: 0

Ilya Gutkovskiy, Jan 22 2018

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Partial sums of A073592.

Euler transform of sequence [-2, -3, -4, -5, -6, -7, -8, -9, ...].

N. J. A. Sloane, Transforms
Index entries for related partition-counting sequences

Cf. A073592, A078616, A191659.

nmax = 48; CoefficientList[Series[Product[(1 - x^k)^k, {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=2} (1 - x^k)^k.

A300521 Expansion of Product_{k>=1} (1 - x^prime(k))^prime(k).

Original entry on oeis.org

1, 0, -2, -3, 1, 1, 3, 0, 9, 8, 4, -31, -12, -13, 20, -13, 48, -17, 74, -87, 8, -143, 175, -174, 349, -164, 369, -651, 520, -1004, 1142, -1218, 1652, -1739, 3291, -3933, 3546, -5743, 6170, -8022, 11435, -13230, 17196, -18706, 22958, -31884, 38420, -49802, 58916

Offset: 0

Ilya Gutkovskiy, Mar 08 2018

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Cf. A000040, A000607, A005063, A007441, A030009, A046675, A073592, A219224, A291647, A291696.

nmax = 48; CoefficientList[Series[Product[(1 - x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[-Sum[DivisorSum[k, Boole[PrimeQ[#]] #^2 &] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^A000040(k))^A000040(k).

G.f.: exp(-Sum_{k>=1} A005063(k)*x^k/k).

cp's OEIS Frontend

A291696 Expansion of Product_{k>=1} (1-x^(k^2))^(k^2).

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A294580 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j)^j.

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A298986 a(n) = [x^n] Product_{k>=1} (1 - n*x^k)^k.

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A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2*n)*x^n/n ) in powers of x.

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A283243 Expansion of exp( Sum_{n>=1} -sigma_2(3*n)*x^n/n ) in powers of x.

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A305205 a(n) = [x^n] exp(-Sum_{k>=1} x^k/(k*(1 - x^k)^n)).

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A299019 Expansion of Product_{k>=1} (1 - x^k)^(k+1).

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A258412 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^k dx.

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A298599 Expansion of Product_{k>=2} (1 - x^k)^k.

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Mathematica

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A300521 Expansion of Product_{k>=1} (1 - x^prime(k))^prime(k).

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A283242 Expansion of exp( Sum_{n>=1} -sigma_2(2n)x^n/n ) in powers of x.

A283243 Expansion of exp( Sum_{n>=1} -sigma_2(3n)x^n/n ) in powers of x.