A008485 -id:A008485 - cp's OEIS frontend

A270915 Decimal expansion of a constant related to the asymptotics of A008485.

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

Offset: 1

Vaclav Kotesovec, Mar 25 2016

			5.352701333486642687772415814165327879851483271286947097319690756...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A008485, A109084, A109085, A206228, A270914, A303070, A304625, A327279.

RealDigits[1/r /. FindRoot[{s == 1/QPochhammer[r*s], QPochhammer[r*s] + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A008485(n)^(1/n).

A309986 Convolution of A270913 and A008485.

Original entry on oeis.org

1, 2, 9, 43, 206, 999, 4915, 24372, 121698, 611244, 3085612, 15645347, 79639602, 406809249, 2084567381, 10712007629, 55187254157, 284981396231, 1474729519719, 7646180479889, 39713643612380, 206600871071930, 1076372569004514, 5615363541987786, 29331204404385053

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

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

Cf. A008485, A270913, A270919, A327214, A327215.

A270913[n_] := SeriesCoefficient[Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}];
A008485[n_] := SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A270913[n-k], {k, 0, n}], {n, 0, 20}]

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687... and c = 0.446705640528056029457240607298917821281915554...

A327215 Self-convolution of A008485.

Original entry on oeis.org

1, 2, 11, 54, 279, 1442, 7530, 39474, 207693, 1095522, 5790116, 30650038, 162451560, 861920492, 4577055823, 24323292984, 129338944225, 688128700422, 3662798123481, 19504467792378, 103899170100154, 553642311668244, 2951010332435759, 15733439067954134

Offset: 0

Vaclav Kotesovec, Aug 26 2019

nonn

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

Cf. A000712, A008485, A304444, A309986, A327214.

A008485[n_]:=SeriesCoefficient[Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A008485[n-k], {k, 0, n}], {n, 0, 25}]

a(n) ~ c^2 * Pi * d^n, where d = A270915 = 5.352701333486642687772415814165... and c = A327279 = 0.26801521271073331568695383828... (see A008485).

A327279 Decimal expansion of a constant related to A008485 and A327215.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 28 2019

			0.26801521271073331568695383828032867950066675724203942644590415846953909499267...

Cf. A008485, A270915, A327215, A327280, A366022.

val = Sqrt[(1 - r*s)*(Log[r*s]^2/(2*Pi*(4*ArcTanh[1 - 2*r*s]*(r*s + (-1 + r*s)*Log[r*s]) - 2*(1 + (-1 + r*s)*ArcTanh[1 - 2*r*s])*Log[1 - r*s] + (-1 + r*s)*(2 + 3*Log[r*s] - 2*Log[1 - r*s]) * QPolyGamma[0, 1, r*s] + (1 - r*s)* QPolyGamma[0, 1, r*s]^2 + (-1 + r*s)*(QPolyGamma[1, 1, r*s] + r*s*Log[r*s]*(r*s^2*Log[r*s] * Derivative[0, 2][QPochhammer][r*s, r*s] - 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s])))))] /. FindRoot[{QPochhammer[r*s] == 1/s, 1/s + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/(s*Log[r*s])}, {r, 1/5}, {s, 2}, WorkingPrecision -> 1000]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]] (* Vaclav Kotesovec, Oct 02 2023 *)

Equals limit_{n->infinity} A008485(n) * sqrt(n) / A270915^n.

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 7, 0, 1, 6, 20, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0

Offset: 0

Alois P. Heinz, Sep 09 2008

A(n,k) is also the number of partitions of n into parts of k kinds.
In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - Geoffrey Critzer, Nov 11 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5, ...
  0,   2,   5,   9,  14,  20, ...
  0,   3,  10,  22,  40,  65, ...
  0,   5,  20,  51, 105, 190, ...
  0,   7,  36, 108, 252, 506, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1.
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. 8.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.
Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).
Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.
Main diagonal gives A008485.
Antidiagonal sums give A067687.
Cf. A060642, A122768, A246935, A255961, A261718.

# DedekindEta is defined in A000594.
A144064Column(k, len) = DedekindEta(len, -k)
for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[0, ] = 1; a[, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]} ]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ M. F. Hasler, Aug 04 2024

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

A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):

A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

Original entry on oeis.org

1, 1, 3, 13, 51, 206, 855, 3585, 15155, 64525, 276278, 1188353, 5130999, 22226049, 96544003, 420368858, 1834203955, 8018057345, 35107961175, 153950675585, 675978772326, 2971700764941, 13078268135683, 57613905606273, 254038914924791, 1121081799217231

Offset: 0

Vaclav Kotesovec, Mar 25 2016

nonn

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p. (End)

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

Cf. A000009, A008485, A255526, A270914, A270917, A270919, A324595, A362408.

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
g:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, b(n),
       (q-> add(g(j, q)*g(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> g(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 31 2021

Table[SeriesCoefficient[Product[(1+x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[QPochhammer[-1, x]^n, {x, 0, n}]/2^n, {n, 0, 25}]
Table[SeriesCoefficient[Exp[n*Sum[(-1)^j*x^j/(j*(x^j - 1)), {j, 1, n}]], {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1, n, (1 + x^k +x*O(x^n))^n), n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 26 2019

a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = A327280 = 0.260542233142438469433860832160...

A109085 G.f. A(x) satisfies: A(x) = P(x*A(x)) where P(x) = A(x/P(x)) is the g.f. of the partition numbers A000041.

Original entry on oeis.org

1, 1, 3, 10, 38, 153, 646, 2816, 12585, 57343, 265401, 1244256, 5896512, 28200365, 135935424, 659754072, 3221354296, 15812501100, 77985955410, 386254209762, 1920391362054, 9580985321554, 47951223856445, 240680464689600

Offset: 0

Paul D. Hanna, Jun 18 2005

nonn

a(n) = Sum[Product(1 + n/h(v)^2)]/(n+1), where the product is over all boxes v in the Ferrers diagram of a partition L of n, h(v) is the hook length of v and the summation is over all partitions L of n. Example: a(3)=10 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, the products are (1+3/9)(1+3/4)(1+3/1)=28/3, (1+3/9)(1+3/1)(1+3/1)=64/3, (1+3/9)(1+3/4)(1+3/1)=28/3 and now a(3)=(1/4)(28+64+28)/3=10. - Emeric Deutsch, May 15 2008

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + ...
G.f. satisfies: P(x*A(x)) = A(x) where P(x) is the partition function:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
The g.f. A = A(x) also satisfies the identities:
(1) A(x) = 1/((1-x*A) * (1-x^2*A^2) * (1-x^3*A^3) * (1-x^4*A^4) * ...).
(2) A(x) = 1 + x*A/(1-x*A) + x^2*A^2/((1-x*A)*(1-x^2*A^2)) + x^3*A^3/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3)) + ...
(3) A(x) = 1 + x*A/(1-x*A)^2 + x^4*A^4/((1-x*A)*(1-x^2*A^2))^2 + x^9*A^9/((1-x*A)*(1-x^2*A^2)*(1-x^3*A^3))^2 + ...
The logarithm of the g.f. is given by:
log(A(x)) = x*A(x)/(1-x*A(x)) + x^2*A(x)^2/(2*(1-x^2*A(x)^2)) + x^3*A(x)^3/(3*(1-x^3*A(x)^3)) + x^4*A(x)^4/(4*(1-x^4*A(x)^4)) + x^5*A(x)^5/(5*(1-x^5*A(x)^5)) + ...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 105*x^4/4 + 506*x^5/5 + 2492*x^6/6 + 12405*x^7/7 + 62337*x^8/8 + ... + A008485(n)*x^n/n + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Guo-Niu Han, An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849 [math.CO]; see p.5 and p.32
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications,arXiv:0805.1398v1 [math.CO], see p.5
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. A109084, A184362, A008485, A000041; related: A106336.

A109085list[n_] := Module[{m = 1, A = 1 + x}, For[i = 1, i <= n, i++, A = 1/Product[(1 - x^k*(A + x*O[x]^n)^k), {k, 1, n}]]; CoefficientList[A, x][[1 ;; n]]]; A109085list[24] (* Jean-François Alcover, Apr 21 2016, adapted from PARI *)
InverseSeries[x QPochhammer[x] + O[x]^25][[3]] (* Vladimir Reshetnikov, Nov 17 2016 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^(n+1), {x, 0, n}]/(n+1), {n, 0, 24}] (* Vladimir Reshetnikov, Nov 20 2016 *)

{a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x); for(i=1, n, A=1/prod(k=1, n, (1-x^k*(A+x*O(x^n))^k))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0,n,x^m*A^m/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0,sqrtint(n+1),(x*A)^(m^2)/prod(k=1, m, (1-x^k*(A+x*O(x^n))^k)^2))); polcoeff(A, n)} \\ Paul D. Hanna, Nov 24 2012
for(n=0,30,print1(a(n),", "))

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(x^m*A^m/m)/(1-x^m*A^m+x*O(x^n)) )));polcoeff(A,n)} \\ Paul D. Hanna, Jun 01 2011

{A008485(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}
{a(n)=polcoeff(exp(sum(m=1,n,A008485(m)*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Feb 06 2012

G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x*eta(x)), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.
(2) A(x) = 1/G(x) where G(x) is g.f. of A109084.
(3) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^n).
(4) A(x) = Sum_{n>=0} x^n*A(x)^n / Product_{k=1..n} (1-x^k*A(x)^k).
(5) A(x) = Sum_{n>=0} (x*A(x))^(n^2) / Product_{k=1..n} (1-x^k*A(x)^k)^2.
(6) A(x) = exp( Sum_{n>=1} (x^n/n) * A(x)^n/(1 - x^n*A(x)^n) ). - Paul D. Hanna, Jun 01 2011
Logarithmic derivative yields A008485, where A008485(n) is the number of partitions of n into parts of n kinds. - Paul D. Hanna, Feb 06 2012
a(n) = ([x^n] 1/((x; x)inf)^(n+1))/(n+1), where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(n) ~ c * d^n / n^(3/2), where d = A270915 = 5.3527013334866426877724... and c = A366022 = 0.489635226684303373081541660578468619322416625... . - Vaclav Kotesovec, Nov 21 2016

A270919 Coefficient of x^n in Product_{k>=1} ((1 + x^k) / (1 - x^k))^n.

Original entry on oeis.org

1, 2, 12, 80, 552, 3912, 28224, 206208, 1520784, 11297546, 84413912, 633713808, 4776117216, 36115518376, 273868321536, 2081866609920, 15859616674336, 121046064563376, 925411686479820, 7085465166635440, 54323193841192752, 416993869451825424, 3204447137019290944

Offset: 0

Vaclav Kotesovec, Mar 25 2016

nonn

From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == 2*p + 2 (mod p^2) holds for all primes p. Cf. A291697. (End)

Seiichi Manyama, Table of n, a(n) for n = 0..1118 (terms 0..500 from Vaclav Kotesovec)

Main diagonal of A288515.

Cf. A008485, A008705, A156616, A270913, A270923, A270924, A291697, A309986, A362408.

Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[(QPochhammer[-1, x]/QPochhammer[x, x])^n, {x, 0, n}]/2^n, {n, 0, 25}]
(* Calculation of constants {d,c}: *) eq = FindRoot[{2*s*QPochhammer[r*s] == QPochhammer[-1, r*s], (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/Log[r*s] + r*((Derivative[0, 1][QPochhammer][-1, r*s] - 2*s*Derivative[0, 1][QPochhammer][r*s, r*s]) / (2*QPochhammer[r*s])) == 1}, {r, 1/8}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[(1 - r*s)*Log[r*s]^2*(QPochhammer[r*s] / (Pi*(-r*s*(-1 + r*s) * Log[r*s]*(4*(2*ArcTanh[1 - 2*r*s] + QPolyGamma[0, 1, r*s])* Derivative[0, 1][QPochhammer][r*s, r*s] + r*Log[r*s]*(Derivative[0, 2][QPochhammer][-1, r*s] - 2*s*Derivative[0, 2][QPochhammer][r*s, r*s])) + 2*QPochhammer[r*s] * (4*r*s*ArcTanh[1 - 2*r*s] + 2*(-1 + (-1 + r*s)*ArcTanh[1 - 2*r*s])*Log[1 - r*s] - (-1 + r*s)*(-2 + Log[r*s] - 2*Log[1 - r*s])*QPolyGamma[0, 1, r*s] + (-1 + r*s) * QPolyGamma[0, 1, r*s]^2 + (-1 + r*s)*(QPolyGamma[1, 1, r*s] - 2*r*s*Log[r*s]* Derivative[0, 0, 1][QPolyGamma][0, 1, r*s])))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ c * d^n / sqrt(n), where d = 7.862983395705905261519347909953827161057584... and c = 0.299856802806668079413694689903953367699319...

a(n) = [x^n] 1/theta_4(x)^n, where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Nov 03 2017

A255672 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(k*n).

Original entry on oeis.org

1, 1, 7, 37, 215, 1251, 7459, 44885, 272727, 1668313, 10263057, 63423482, 393440867, 2448542136, 15280435191, 95588065737, 599213418327, 3763242239317, 23673166664695, 149138199543613, 940796936557265, 5941862248557566, 37568309060087582, 237767215209245583

Offset: 0

Vaclav Kotesovec, Mar 01 2015

nonn

Number of partitions of n when parts i are of n*i kinds. - Alois P. Heinz, Nov 23 2018
From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 3 and all positive integers n and k. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from Vaclav Kotesovec)

Cf. A008485, A252782, A270913, A270922, A380290.

Main diagonal of A255961.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

Table[SeriesCoefficient[Product[1/(1-x^k)^(k*n),{k,1,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 01 2015 *)

a(n) ~ c * d^n / sqrt(n), where d = 6.468409145117839606941857350154192468889057616577..., c = 0.25864792865819067933968646380369970564... . - Vaclav Kotesovec, Mar 01 2015

a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A255526 Coefficient of x^n in Product_{k>=1} 1/(1+x^k)^n.

Original entry on oeis.org

-1, 1, -4, 17, -56, 172, -547, 1809, -6061, 20316, -68135, 229244, -774372, 2624119, -8912759, 30328593, -103382254, 352975681, -1206921212, 4132159452, -14163858895, 48601267199, -166930975524, 573872089212, -1974472043081, 6798561779868, -23425506369715

Offset: 1

Vaclav Kotesovec, Feb 24 2015

sign

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A008485, A270913, A278428.

Table[SeriesCoefficient[Product[1/(1+x^k)^n,{k,1,n}],{x,0,n}],{n,1,30}]
(* Calculation of constant c: *) 1/Sqrt[(4 - r^2*s^3*Derivative[0, 2][QPochhammer][-1, r*s])*Pi] /. FindRoot[{QPochhammer[-1, r*s] == 2/s, 2/s + r*s*Derivative[0, 1][QPochhammer][-1, r*s] == 0}, {r, -1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ (-1)^n * c * d^n / sqrt(n), where d = A318204 = 3.5097543279497033404372735..., c = 0.23322106096789389697797... .

cp's OEIS Frontend

A270915 Decimal expansion of a constant related to the asymptotics of A008485.

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A309986 Convolution of A270913 and A008485.

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A327215 Self-convolution of A008485.

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A327279 Decimal expansion of a constant related to A008485 and A327215.

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A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

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A270913 Coefficient of x^n in Product_{k>=1} (1+x^k)^n.

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A109085 G.f. A(x) satisfies: A(x) = P(x*A(x)) where P(x) = A(x/P(x)) is the g.f. of the partition numbers A000041.

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