A144048 -id:A144048 - cp's OEIS frontend

A023875 Expansion of Product_{k>=1} (1 - x^k)^(-k^6).

1, 1, 65, 794, 6970, 69251, 689896, 6309849, 55654858, 483526120, 4104495070, 33968248260, 275366110929, 2192975727284, 17169583920204, 132264358228507, 1003715206329332, 7511468689508580, 55479733165442038, 404709688656248024, 2917717129031507178

Offset: 0

Olivier Gérard

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1771 (first 451 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Vaclav Kotesovec, Graph - The asymptotic ratio for 10000 terms
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

Column k=6 of A144048.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^6: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^6, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 02 2012

max = 20; Series[ Product[1/(1 - x^k)^k^6, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* Jean-François Alcover, Mar 05 2013 *)

m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^6)) \\ G. C. Greubel, Oct 31 2018

a(n) ~ exp(Pi * 2^(27/8) * n^(7/8) / (7*15^(1/8)) - 45*Zeta(7) / (8*Pi^6)) / (2^(29/16) * 15^(1/16) * n^(9/16)), where Zeta(7) = A013665 = 1.00834927738192... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp( Sum_{n>=1} sigma_7(n)*x^n/n ). - Seiichi Manyama, Mar 05 2017
a(n) = (1/n)*Sum_{k=1..n} sigma_7(k)*a(n-k). - Seiichi Manyama, Mar 05 2017

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A023876 G.f.: Product_{k>=1} (1 - x^k)^(-k^7).

Original entry on oeis.org

1, 1, 129, 2316, 26956, 385017, 5512443, 70223666, 866470849, 10628564312, 126832407040, 1469751196093, 16694372607012, 186350644088784, 2042610304126944, 22007441766651756, 233482509248479425, 2441727926157182541, 25187101530316996950, 256456174925807404269

Offset: 0

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Vaclav Kotesovec, Graph - The asymptotic ratio for 10000 terms
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

Column k=7 of A144048.

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^7: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^7, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 02 2012

max = 19; Series[ Product[1/(1 - x^k)^k^7, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* Jean-François Alcover, Mar 05 2013 *)

m=20; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^7)) \\ G. C. Greubel, Oct 31 2018

a(n) ~ (35*Zeta(9))^(119/2160) * exp((3/2)^(20/9) * n^(8/9) * (35*Zeta(9))^(1/9) + Zeta'(-7)) / (2^(247/2160) * 3^(961/1080) * sqrt(Pi) * n^(1199/2160)), where Zeta(9) = A013667 = 1.0020083928260822144..., Zeta'(-7) = ((gamma + log(2*Pi) - 363/140)/30 - 315*Zeta'(8)/Pi^8)/8 = -0.00072864268015924... . - Vaclav Kotesovec, Feb 27 2015

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A023877 Expansion of Product_{k>=1} (1 - x^k)^(-k^8).

Original entry on oeis.org

1, 1, 257, 6818, 105250, 2175491, 44988020, 796565173, 13803604854, 240522266760, 4044067171130, 65769795259820, 1051279656603367, 16517653032316394, 254354069377336990, 3847172021760617755, 57300325471166205776, 840900188345961238222, 12164188625099191500782

Offset: 0

Olivier Gérard

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1170 (first 301 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 21.

Column k=8 of A144048.

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^8: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^8, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Nov 02 2012

max = 18; Series[ Product[1/(1 - x^k)^k^8, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x] & (* Jean-François Alcover, Mar 05 2013 *)

m=20; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^8)) \\ G. C. Greubel, Oct 31 2018

a(n) ~ exp(5 * Pi * 2^(17/10) * n^(9/10) / (3^(21/10) * 11^(1/10)) + 315*Zeta(9)/(4*Pi^8)) / (2^(13/20) * sqrt(5) * 33^(1/20) * n^(11/20)), where Zeta(9) = A013667 = 1.0020083928260822144... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp( Sum_{n>=1} sigma_9(n)*x^n/n ). - Seiichi Manyama, Mar 05 2017
a(n) = (1/n)*Sum_{k=1..n} sigma_9(k)*a(n-k). - Seiichi Manyama, Mar 05 2017

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A284993 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j^k) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -3, -2, 1, 1, -1, -7, -6, 1, -1, 1, -1, -15, -20, 0, 0, 1, 1, -1, -31, -66, -8, 11, 4, -1, 1, -1, -63, -212, -54, 99, 42, 2, 2, 1, -1, -127, -666, -284, 725, 455, 63, 8, -2, 1, -1, -255, -2060, -1350, 4935, 4580, 958, 73

Offset: 0

Seiichi Manyama, Apr 07 2017

			Square array begins:
   1,  1,  1,   1,   1,    1, ...
  -1, -1, -1,  -1,  -1,   -1, ...
   0, -1, -3,  -7, -15,  -31, ...
  -1, -2, -6, -20, -66, -212, ...
   1,  1,  0,  -8, -54, -284, ...

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

Columns k=0-5 give A081362, A255528, A284896, A284897, A284898, A284899.

Cf. A144048, A284992.

G.f. of column k: Product_{j>=1} 1/(1+x^j)^(j^k).

A294296 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} sigma_k(j) * x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 7, 25, 1, 1, 11, 43, 193, 1, 1, 19, 91, 409, 1481, 1, 1, 35, 223, 1105, 3841, 16021, 1, 1, 67, 595, 3505, 13841, 50431, 167665, 1, 1, 131, 1663, 12193, 60841, 230731, 648187, 2220065, 1, 1, 259, 4771, 44689, 297761, 1340851, 3955771

Offset: 0

Seiichi Manyama, Oct 30 2017

			Square array A(n,k) begins:
      1,    1,     1,     1,      1, ...
      1,    1,     1,     1,      1, ...
      5,    7,    11,    19,     35, ...
     25,   43,    91,   223,    595, ...
    193,  409,  1105,  3505,  12193, ...
   1481, 3841, 13841, 60841, 297761, ...

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

Columns k=0..2 give A294363, A294361, A294362.
Rows n=0-1 give A000012.
Main diagonal gives A294388.
Cf. A144048.

A(0,k) = 1 and A(n,k) = (n-1)! * Sum_{j=1..n} j*sigma_k(j)*A(n-j,k)/(n-j)! for n > 0.

A294589 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^(j^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 9, 14, 14, 1, 1, 17, 36, 42, 25, 1, 1, 33, 98, 140, 103, 56, 1, 1, 65, 276, 498, 481, 289, 97, 1, 1, 129, 794, 1844, 2419, 1774, 690, 198, 1, 1, 257, 2316, 7002, 12745, 12173, 5925, 1771, 354, 1, 1, 513, 6818, 27020, 69283, 89706, 56974, 20076, 4206, 672, 1

Offset: 0

Seiichi Manyama, Nov 03 2017

			Square array begins:
    1,  1,   1,   1,    1, ...
    1,  1,   1,   1,    1, ...
    3,  5,   9,  17,   33, ...
    6, 14,  36,  98,  276, ...
   14, 42, 140, 498, 1844, ...

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

Columns k=0..3 give A006906, A266941, A285241, A294590.
Rows n=0-1 give A000012.
Cf. A144048, A294587.

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.

A319361 a(n) = [x^n] exp(Sum_{k>=1} sigma_n(k)*x^k/k).

Original entry on oeis.org

1, 1, 3, 14, 136, 2411, 88903, 6309849, 866470849, 240522266760, 132000248840652, 141226630324344532, 306101744973083495408, 1327520858367342045830198, 11328405846086223895036194126, 196814026990537767059856457640779, 6894163531963490274906095710739747873

Offset: 0

Ilya Gutkovskiy, Sep 17 2018

nonn

Diagonal of A144048.

Cf. A252782, A305207.

Table[SeriesCoefficient[Exp[Sum[DivisorSigma[n, k] x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 16}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^(k^(n - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 16}]

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

A320251 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{j>=1} j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 5, 8, 8, 1, 1, 9, 18, 21, 16, 1, 1, 17, 44, 63, 55, 32, 1, 1, 33, 114, 207, 221, 144, 64, 1, 1, 65, 308, 723, 991, 776, 377, 128, 1, 1, 129, 858, 2631, 4805, 4752, 2725, 987, 256, 1, 1, 257, 2444, 9843, 24655, 31880, 22769, 9569, 2584, 512

Offset: 0

Ilya Gutkovskiy, Oct 08 2018

A(n,k) is the invert transform of k-th powers evaluated at n.

			G.f. of column k: A_k(x) = 1 + x + (2^k + 1)*x^2 + (2^(k + 1) + 3^k + 1)*x^3 + (3*2^k + 2^(2*k + 1) + 2*3^k + 1)*x^4 + ...
Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   1,   1,    1,    1,     1,      1,  ...
   2,   3,    5,    9,    17,     33,  ...
   4,   8,   18,   44,   114,    308,  ...
   8,  21,   63,  207,   723,   2631,  ...
  16,  55,  221,  991,  4805,  24655,  ...

N. J. A. Sloane, Transforms

Columns k=0..3 give A011782, A088305, A033453, A144109.
Main diagonal gives A301655.
Cf. A144048.

Table[Function[k, SeriesCoefficient[1/(1 - Sum[i^k x^i, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - PolyLog[-k, x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 - PolyLog(-k,x)), where PolyLog() is the polylogarithm function.

cp's OEIS Frontend

A023875 Expansion of Product_{k>=1} (1 - x^k)^(-k^6).

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A023876 G.f.: Product_{k>=1} (1 - x^k)^(-k^7).

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A023877 Expansion of Product_{k>=1} (1 - x^k)^(-k^8).

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A284993 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1+x^j)^(j^k) in powers of x.

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A294296 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f.: exp(Sum_{j>=1} sigma_k(j) * x^j).

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A294589 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j*x^j)^(j^k).

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A319361 a(n) = [x^n] exp(Sum_{k>=1} sigma_n(k)*x^k/k).

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A320251 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{j>=1} j^k*x^j).

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