A000728 -id:A000728 - cp's OEIS frontend

A282937 a(n) = A000728(5*n).

1, -6, 9, 10, -30, 1, 5, 51, 10, -100, 20, -55, 109, 110, -130, -1, -110, 160, 10, -230, 100, 15, 191, 120, -230, -100, -89, 160, 90, -340, 120, 5, 300, 200, -260, -1, -275, 240, -100, -270, 119, -165, 260, 410, -200, -40, 20, 200, -110, -500, 180, -54, 140

Offset: 0

Seiichi Manyama, Feb 25 2017

sign

This generalized function is related to two following identities; R(q^5) - q - q^2/R(q^5) = (q; q){infinity}/(q^25; q^25){infinity}, R^5(q^5) - 11*q^5 - q^10/R^5(q^5) = ((q^5; q^5){infinity}/(q^25; q^25){infinity})^6, where R(q) is the Rogers-Ramanujan continued function and (q; q)_n is the q-Pochhammer symbol. See the reference.

			G.f.: 1 - 6*q + 9*q^2 + 10*q^3 - 30*q^4 + q^5 + 5*q^6 + 51*q^7 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part III, Springer, New York, 2012, See p. 185.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Cf. A000728, A182821 (Product_{n>=1} (1 - q^(5*n))/(1 - q^n)^6), A282941.

Cf. Product_{n>=1} (1 - q^n)^(k+1)/(1 - q^(k*n)): A010815 (k=1), A115110 (k=2), A185654 (k=3), this sequence (k=5), A282942 (k=7).

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^j)^6/(1 - x^(5*j)): j in [1..(m+2)]]) )); // G. C. Greubel, Nov 18 2018

CoefficientList[Series[Product[(1 - x^j)^6/(1 - x^(5*j)), {j,1,62}], {x,0,60}], x] (* G. C. Greubel, Nov 18 2018 *)

m=60; x='x+O('x^m); Vec(prod(j=1,m+2, (1 - x^j)^6/(1 - x^(5*j)))) \\ G. C. Greubel, Nov 18 2018

R = PowerSeriesRing(ZZ, 'x')
prec = 60
x = R.gen().O(prec)
s = prod((1 - x^j)^6/(1 - x^(5*j)) for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

G.f.: Product_{n>=1} (1 - q^n)^6/(1 - q^(5*n)).
a(n) = (-1)^j mod 5 if n = j*(3*j - 1)/2 for all j in Z; otherwise a(n) = 0 mod 5.
Sum_{k=0..n} a(k)*A182821(n-k) = 0 for n > 0. - Seiichi Manyama, Feb 28 2017
G.f.: exp( Sum_{n>=1} -sigma(5*n)*q^n/n ). - Seiichi Manyama, Mar 04 2017
a(n) = -(1/n)*Sum_{k=1..n} sigma(5*k)*a(n-k). - Seiichi Manyama, Mar 04 2017

A286354 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, -1, 0, 1, -3, -1, 0, 0, 1, -4, 0, 2, 0, 0, 1, -5, 2, 5, 1, 1, 0, 1, -6, 5, 8, 0, 2, 0, 0, 1, -7, 9, 10, -5, 0, -2, 1, 0, 1, -8, 14, 10, -15, -4, -7, 0, 0, 0, 1, -9, 20, 7, -30, -6, -10, 0, -2, 0, 0, 1, -10, 27, 0, -49, 0, -5, 8, 0, -2, 0, 0, 1, -11, 35, -12, -70, 21, 11, 25, 9, 0, 1, 0, 0

Offset: 0

Ilya Gutkovskiy, May 08 2017

A(n,k) number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts with k types of each part.

			A(3,2) = 2 because we have [2, 1], [2', 1], [2, 1'], [2', 1'] (number of partitions of 3 into an even number of distinct parts with 2 types of each part), [3], [3'] (number of partitions of 3 into an odd number of distinct parts with 2 types of each part) and 4 - 2 = 2.
Square array begins:
1,  1,  1,  1,  1,   1,  ...
0, -1, -2, -3, -4,  -5,  ...
0, -1, -1,  0,  2,   5,  ...
0,  0,  2,  5,  8,  10,  ...
0,  0,  1,  0, -5, -15,  ...
0,  1,  2,  0, -4,  -6,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-20 give: A000007, A010815, A002107, A010816, A000727, A000728, A000729, A000730, A000731, A010817, A010818, A010819, A000735, A010820, A010821, A010822, A000739, A010823, A010824, A010825, A010826.
Main diagonal gives A008705.
Antidiagonal sums give A299105.

A:= proc(n, k) option remember; `if`(n=0, 1, -k*
      add(numtheory[sigma](j)*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jun 21 2018

Table[Function[k, SeriesCoefficient[Product[(1 - x^i)^k , {i, Infinity}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[x, x, Infinity]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[(-1)^i*x^(i*(3*i + 1)/2), {i, -Infinity, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 - x^j)^k.
G.f. of column k: (Sum_{j=-inf..inf} (-1)^j*x^(j*(3*j+1)/2))^k.
Column k is the Euler transform of period 1 sequence [-k, -k, -k, ...].

A302057 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^5 is zero.

Original entry on oeis.org

1560, 1802, 1838, 2318, 2690, 3174, 3742, 3925, 4348, 4710, 4854, 5002, 5092, 5210, 7484, 7615, 8796, 8846, 9500, 10345, 12110, 14178, 14972, 16203, 18010, 19314, 20207, 20406, 20679, 24566, 25231, 27403, 27532, 28361, 31567, 31573, 35610, 35795, 37347

Offset: 1

Ilya Gutkovskiy, Mar 31 2018

nonn

Numbers k such that number of partitions of k into an even number of distinct parts equals number of partitions of k into an odd number of distinct parts, with 5 types of each part.

Joerg Arndt, Table of n, a(n) for n = 1..1212 (terms 1..53 from Seiichi Manyama, terms 54..81 from Jean-François Alcover)
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m = 1), A213250 (m = 2), A014132 (m = 3), A302056 (m = 4), this sequence (m = 5), A020757 (m = 6), A322043 (m = 15).

Cf. A000728.

Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[QPochhammer[x]^5, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[Sum[(-1)^j x^(j (3 j + 1)/2), {j, -nmax, nmax}]^5, {x, 0, nmax}], x]], 0]]
Flatten[Position[nmax = 38000; Rest[CoefficientList[Series[Exp[-5 Sum[DivisorSigma[1, j] x^j/j, {j, 1, nmax}]], {x, 0, nmax}], x]], 0]]
(* 4th program: *)
sigma[k_] := sigma[k] = DivisorSigma[1, k];
a[0] = 1; a[n_] := a[n] = -5/n Sum[sigma[k] a[n-k], {k, 1, n}];
Reap[For[k = 1, k <= 10^5, k++, If[a[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Dec 20 2018 *)

x='x+O('x^30000); v=Vec(eta(x)^5 - 1); for(k=1, #v, if(v[k]==0, print1(k, ", "))); \\ Altug Alkan, Mar 31 2018, after Joerg Arndt at A213250

A339704 Dirichlet g.f.: Product_{k>=2} (1 - k^(-s))^5.

Original entry on oeis.org

1, -5, -5, 5, -5, 20, -5, 10, 5, 20, -5, -5, -5, 20, 20, -15, -5, -5, -5, -5, 20, 20, -5, -55, 5, 20, 10, -5, -5, -55, -5, -6, 20, 20, 20, -45, -5, 20, 20, -55, -5, -55, -5, -5, -5, 20, -5, 20, 5, -5, 20, -5, -5, -55, 20, -55, 20, 20, -5, -55, -5, 20, -5, -5, 20, -55, -5, -5, 20, -55

Offset: 1

Ilya Gutkovskiy, Dec 13 2020

sign

Cf. A000728, A114592, A339320, A339701, A339702, A339703, A339705, A339706, A339707.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339320(n/d) * a(d).

a(p^k) = A000728(k) for prime p.

A258405 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^5 dx.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 29 2015

			0.137801070846554642845386131402193843084522541232625982683937003709248631...

Vaclav Kotesovec, The integration of q-series

Cf. A000728, A258232, A258406, A258407, A258404, A258405.

evalf(Sum(Sum(2*Pi*(-1)^(h+m) / cosh(sqrt(7 - 4*h + 12*h^2 - 4*m + 12*m^2)*Pi/2), m=-infinity..infinity), h=-infinity..infinity), 120); # Vaclav Kotesovec, Dec 04 2015

nmax=200; p=1; q5=Table[PrintTemporary[n]; p=Expand[p*(1-x^n)^5]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]],{n,1,nmax}]; q5n=N[q5,1000]; Table[SequenceLimit[Take[q5n,j]],{j,Length[q5n]-100,Length[q5n],10}]
nterms = 40; N[Sum[Sum[2*Pi*(-1)^(h+m) / Cosh[Sqrt[7 - 4*h + 12*h^2 - 4*m + 12*m^2]*Pi/2], {m, -nterms, nterms}], {h, -nterms, nterms}], 100] (* Vaclav Kotesovec, Dec 04 2015 *)
RealDigits[NIntegrate[QPochhammer[x]^5, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

Sum_{m = -infinity..infinity} (Sum_{h = -infinity..infinity} (2*Pi*(-1)^(h+m) / cosh(sqrt(7 - 4*h + 12*h^2 - 4*m + 12*m^2)*Pi/2))). - Vaclav Kotesovec, Dec 04 2015

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

cp's OEIS Frontend

A282937 a(n) = A000728(5*n).

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A286354 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)^k.

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A302057 Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^5 is zero.

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A339704 Dirichlet g.f.: Product_{k>=2} (1 - k^(-s))^5.

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A258405 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^5 dx.

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A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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