A073592 -id:A073592 - cp's OEIS frontend

A283338 Expansion of exp( Sum_{n>=1} -sigma_8(n)*x^n/n ) in powers of x.

1, -1, -128, -2059, -6069, 210067, 3664420, 23366098, -116899962, -4133365357, -41809923367, -125160180169, 2447495850838, 42931762306584, 321967686614676, 281683012498569, -23874414003295851, -318729240693402530, -1992572289343189863

Offset: 0

Seiichi Manyama, Mar 05 2017

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

Column k=7 of A283272.
Cf. A023876 (exp( Sum_{n>=1} sigma_8(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), A283337 (k=7), this sequence (k=8), A283339 (k=9), A283340 (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^7).

a(n) = -(1/n)*Sum_{k=1..n} sigma_8(k)*a(n-k).

A283339 Expansion of exp( Sum_{n>=1} -sigma_9(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -256, -6305, -26335, 1321887, 32565169, 276211695, -2659962750, -111341327890, -1454216029918, -3323783801026, 227018039015019, 4636828146319845, 39615489757794355, -132865771935151820, -9075288352543844755, -132703303201618610765

Offset: 0

Seiichi Manyama, Mar 05 2017

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

Column k=8 of A283272.
Cf. A023877 (exp( Sum_{n>=1} sigma_9(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), A283337 (k=7), A283338 (k=8), this sequence (k=9), A283340 (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^8).

a(n) = -(1/n)*Sum_{k=1..n} sigma_9(k)*a(n-k).

A283340 Expansion of exp( Sum_{n>=1} -sigma_10(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -512, -19171, -111645, 8255899, 287477144, 3248973702, -56353404842, -2946880278857, -50078654012311, -24091665240825, 19437354184565824, 486126425619195338, 4607922953609319032, -63107867988829247005, -3101395214088243725145

Offset: 0

Seiichi Manyama, Mar 05 2017

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

Column k=9 of A283272.
Cf. A023878 (exp( Sum_{n>=1} sigma_10(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), A283271 (k=5), A283336 (k=6), A283337 (k=7), A283338 (k=8), A283339 (k=9), this sequence (k=10).

G.f.: Product_{n>=1} (1 - x^n)^(n^9).

a(n) = -(1/n)*Sum_{k=1..n} sigma_10(k)*a(n-k).

A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

Original entry on oeis.org

1, 1, 0, -3, -6, -4, 12, 39, 52, -9, -186, -392, -285, 610, 2291, 3200, -150, -10626, -23487, -18841, 32957, 134848, 198246, 13961, -605248, -1409604, -1234474, 1744213, 7898753, 12209679, 2161666, -34344627, -84393284, -79993042, 90692470, 461463974, 749309529, 207447895, -1939084232

Offset: 0

Ilya Gutkovskiy, Feb 05 2018

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Robert Israel, Table of n, a(n) for n = 0..3925
N. J. A. Sloane, Transforms

Antidiagonal sums of A276554.

Cf. A067687, A073592, A299105, A299106, A299108, A299162, A299164, A299166, A299167, A299208, A299209, A299210, A299212.

N:= 100: # for a(0)..a(N)
S:= series(1/(1-x*mul((1-x^k)^k,k=1..N)),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Feb 05 2023

nmax = 38; CoefficientList[Series[1/(1 - x Product[(1 - x^k)^k, {k, 1, nmax}]), {x, 0, nmax}], x]

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

a(0) = 1; a(n) = Sum_{k=1..n} A073592(k-1)*a(n-k).

A318784 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

Original entry on oeis.org

1, 0, 1, 1, 4, 2, 11, 6, 25, 20, 56, 44, 139, 107, 283, 266, 619, 567, 1317, 1242, 2680, 2705, 5403, 5539, 10947, 11339, 21291, 23013, 41494, 45213, 79991, 88312, 151546, 170908, 284901, 324421, 532505, 611227, 981002, 1142000, 1797451, 2105773, 3268765, 3855050, 5889704, 7004451

Offset: 0

Ilya Gutkovskiy, Sep 03 2018

nonn

Convolution of A061256 and A073592.

Euler transform of A001065.

N. J. A. Sloane, Transforms

Cf. A000203, A001065, A001157, A061256, A073592, A318783.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      (sigma(d)-d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 03 2018

nmax = 45; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^(2 k)/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A001065(k).
G.f.: exp(Sum_{k>=1} sigma_2(k)*x^(2*k)/(k*(1 - x^k))), where sigma_2(k) = sum of squares of divisors of k (A001157).
a(n) ~ exp(3^(2/3) * c^(1/3) * n^(2/3)/2 - Pi^2 * n^(1/3) / (4 * 3^(2/3) * c^(1/3)) - Pi^4/(288*c) - 1/8) * A^(3/2) * c^(1/8) / (3^(5/8) * (2*Pi)^(11/24) * n^(5/8)), where c = (Pi^2 - 6)*Zeta(3) and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Sep 03 2018

A294404 E.g.f.: exp(-Sum_{n>=1} sigma_2(n) * x^n).

Original entry on oeis.org

1, -1, -9, -31, -23, 3399, 41311, 473129, 1284081, -79051537, -2447228249, -52444297071, -712806368999, -2221410364681, 331443685309647, 15068893004257049, 460836352976093281, 10298306504802529119, 122928784866003823831, -3359583359629857247807

Offset: 0

Seiichi Manyama, Oct 30 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..426

E.g.f.: exp(-Sum_{n>=1} sigma_k(n) * x^n): A294402 (k=0), A294403 (k=1), this sequence (k=2).

Cf. A001157, A073592, A294362, A343497.

N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sigma(k, 2)*x^k))))

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A001157(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} gcd(k,j)^3. - Ilya Gutkovskiy, Aug 17 2021

A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -8, -1, 1, -1, -32, -73, 0, 1, -1, -128, -2155, -927, 4, 1, -1, -512, -58921, -259701, -13969, 4, 1, -1, -2048, -1593811, -67045719, -48496253, -254580, 7, 1, -1, -8192, -43044673, -17178209325, -152513227585, -13001952944, -5288596, 3

Offset: 0

Seiichi Manyama, Nov 09 2017

			Square array begins:
    1,      1,         1,             1,                1, ...
   -1,     -1,        -1,            -1,               -1, ...
   -2,     -8,       -32,          -128,             -512, ...
   -1,    -73,     -2155,        -58921,         -1593811, ...
    0,   -927,   -259701,     -67045719,     -17178209325, ...
    4, -13969, -48496253, -152513227585, -476819162106101, ...

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

Columns k=0..2 give A073592, A294809, A294953.
Rows n=0..2 give A000012, (-1)*A000012, (-1)*A004171.
Cf. A294653, A294950.

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

A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, -1, 4, 1, -17, -6, 118, -8, -876, 625, 5966, -7486, -41937, 75969, 306312, -768637, -2164992, 7487063, 14461466, -70259884, -89410774, 646971980, 459817892, -5861484630, -1128608133, 52082250637, -15894742662, -453574650852, 366848121166, 3866670213663, -5215687717614

Offset: 0

Ilya Gutkovskiy, Mar 24 2018

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			G.f. A(x) = 1 - x - x^2 + 4*x^3 + x^4 - 17*x^5 - 6*x^6 + 118*x^7 - 8*x^8 - 876*x^9 + 625*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * (1 - x^4*A(x)^4)^4 * ...
log(A(x)) = -x - 3*x^2/2 + 8*x^3/3 + 13*x^4/4 - 51*x^5/5 - 120*x^6/6 + 538*x^7/7 + 781*x^8/8 - 5419*x^9/9 - 3053*x^10/10 + ... + A281267(n)*x^n/n + ...

Cf. A000219, A006195, A066398, A073592, A109085, A181315, A278428, A281267, A301455, A301456, A301625.

with(numtheory):
Order := 33:
Gser := solve(series(x*exp(add(sigma[2](n)*x^n/n, n = 1..32)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..32); # Peter Bala, Feb 09 2020

From Peter Bala, Feb 09 2020: (Start)
A(x) = 1/x * series reversion of ( exp( Sum_{n >= 1} sigma_2(n)*x^n/n ) ), where sigma_2(n) = A001157(n).
Equivalently, the o.g.f. A(x) satisfies [x^n](1/A(x))^n = sigma_2(n) for n >= 1. Cf. A066398. (End)
A(x) equals (1/x) * series reversion of (x * the o.g.f. for the sequence of planar partitions A000219).  - Peter Bala, Feb 11 2020

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

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			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}]

A276552 Expansion of Product_{k>0} (1 - x^k)^(k*3).

Original entry on oeis.org

1, -3, -3, 8, 12, 9, -29, -54, -51, 8, 168, 273, 270, -18, -546, -1220, -1539, -969, 796, 3693, 6591, 8098, 5412, -2568, -16053, -31524, -42195, -38684, -11868, 41994, 117630, 193365, 235497, 197758, 42852, -247224, -639547, -1041432, -1291425, -1184100, -520650

Offset: 0

Seiichi Manyama, Apr 10 2017

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

Column k=3 of A276554.

Product_{k>0} (1-x^k)^(k*m): A255610 (m=-3), A073592 (m=1), A276551 (m=2), this sequence (m=3).

cp's OEIS Frontend

A283338 Expansion of exp( Sum_{n>=1} -sigma_8(n)*x^n/n ) in powers of x.

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A283339 Expansion of exp( Sum_{n>=1} -sigma_9(n)*x^n/n ) in powers of x.

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A283340 Expansion of exp( Sum_{n>=1} -sigma_10(n)*x^n/n ) in powers of x.

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A299211 Expansion of 1/(1 - x*Product_{k>=1} (1 - x^k)^k).

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Maple

Mathematica

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A318784 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203).

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Maple

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A294404 E.g.f.: exp(-Sum_{n>=1} sigma_2(n) * x^n).

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PARI

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A294808 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*j)*x^j)^j in powers of x.

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A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

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Maple

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

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Mathematica

A276552 Expansion of Product_{k>0} (1 - x^k)^(k*3).

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A294808 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-j^(kj)x^j)^j in powers of x.