A258407 -id:A258407 - cp's OEIS frontend

A010816 Expansion of Product_{k>=1} (1 - x^k)^3.

1, -3, 0, 5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -27, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Also, number of different partitions of n into parts of -3 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004

			G.f. = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 + 13*x^21 - 15*x^28 + ...
G.f. for b(n): = q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 + ...

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 117, Problem 22.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.14).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003, p. 285, Theorem 357 (Jacobi).
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410, Problem 23.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 267 MR0099904 (20 #6340)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
V. Kotesovec, The integration of q-series
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.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000594, A116916, A258407.

# DedekindEta is defined in A000594.
A010816List(len) = DedekindEta(len, 3)
A010816List(39) |> println # Peter Luschny, Mar 10 2018

S:= series(mul(1-x^k,k=1..200)^3,x,201):
seq(coeff(S,x,j),j=0..200); # Robert Israel, Feb 01 2018
A010816 := n -> if issqr(8*n+1) then isqrt(8*n+1); (-1)^iquo(%, 2) * % else 0 fi:
seq(A010816(n), n=0..98); # Peter Luschny, Apr 17 2022

a[ n_] := SeriesCoefficient[ EllipticThetaPrime[ 1, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* Michael Somos, Oct 22 2011 *)
a[ n_] := With[ {m = 8 n + 1}, If[m > 0 && OddQ[ Length @ Divisors @ m], Sqrt[m] KroneckerSymbol[-4, Sqrt[m]], 0]];  (* Michael Somos, Aug 26 2015 *)
CoefficientList[QPochhammer[q]^3 + O[q]^100, q] (* Jean-François Alcover, Nov 25 2015 *)
a[ n_] := With[ {x = Sqrt[8 n + 1]}, If[ IntegerQ[ x], (-1)^Quotient[ x, 2] x, 0]]; (* Michael Somos, Feb 01 2018 *)
a[ n_] := If[ n < 1, Boole[ n == 0], Times @@ (If[ # == 2 || OddQ[ #2], 0, (KroneckerSymbol[ -4, #] #)^(#2/2)] & @@@ FactorInteger[ 8 n + 1])]; (* Michael Somos, Feb 01 2018 *)

{a(n) = my(x); if( n<0, 0, if( issquare( 8*n + 1, &x), (-1)^(x\2) * x))}; /* Michael Somos, Nov 08 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3, n))};

from sympy import integer_nthroot
def A010816(n):
    a, b = integer_nthroot((n<<3)+1,2)
    return (-a if a&2 else a) if b else 0 # Chai Wah Wu, Nov 02 2024

G.f.: Product_{k>=1} (1-x^k)^3 = Sum_{n>=0} (-1)^n*(2*n+1)*x^(n*(n+1)/2) (Jacobi).
Given g.f. A(x), then q * A(q^8) = eta(q^8)^3 = theta_2(q^4)*theta_3*(q^4)*theta_4(q^4) / 2 = theta_1'(q^4) / (2*Pi). - Michael Somos, Nov 08 2005
Given g.f. A(x), then x*A(x)^8 is g.f. for A000594.
a(n) = b(8*n + 1) where b() is multiplicative with b(p^e) = 0 if e odd, b(2^e) = 0^e, b(p^e) = p^(e/2) if p == 1 (mod 4), b(p^e) = (-p)^(e/2) if p == 3 (mod 4). - Michael Somos, Aug 22 2006
Expansion of f(-x)^3 in powers of x where f() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 2^(9/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 09 2007
a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = -3 * a(n). a(25*n + 3) = 5 * a(n). - Michael Somos, Sep 09 2007
a(3*n) = A116916(n).
a(n) = (t*(t+1)-2*n-1)*(t-r)*(-1)^(t+1), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2). - Mikael Aaltonen, Jan 17 2015
a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017
G.f.: exp(-3*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
G.f.: Product_{n >= 1} (1 - q^(4*n))^3 * (1 + q^(4*n-1))^(-3) * (1 - q^(4*n-2))^6 * (1 + q^(4*n-3))^(-3). - Peter Bala, Jun 07 2025

A258232 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 24 2015

			0.3684125359314336523213165973278510150142413039288199683036158...

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 424.

Jonathan M. Borwein and Peter B. Borwein, Strange series and high precision fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640; alternative link.
Martin Klazar, What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, The integration of q-series.

Cf. A000120, A010815, A029931, A242168, A258229, A258230, A258408.

Cf. A258406 (m=2), A258407 (m=3), A258404 (m=4), A258405 (m=5).

evalf(8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6)/(2*cosh(sqrt(23)*Pi/3)-1), 123);
evalf(Sum((-1)^n/((3*n-1)*n/2 + 1), n=-infinity..infinity), 123);

RealDigits[N[8*Sqrt[3/23]*Pi*Sinh[Sqrt[23]*Pi/6] / (2*Cosh[Sqrt[23]*Pi/3]-1),120]][[1]]

8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1) \\ Michel Marcus, Nov 28 2018

Equals 8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1).
From Amiram Eldar, Feb 04 2024: (Start)
Equals 2 * Sum_{k=-oo..oo} (-1)^k/(3*k^2 + k + 2).
Equals Sum_{k>=0} (-1)^A000120(k)/(A029931(k)+1) (Borwein and Borwein, 1992). (End)

A258406 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^2 dx.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 29 2015

			0.2538740823782760029885088938163329123847636343193313514756067...

Vaclav Kotesovec, The integration of q-series

Cf. A002107, A258232, A258407, A258404, A258405.

evalf(Sum(Sum(8*(n+1)*(-1)^n / ((n^2 - 2*k^2 + 2*k*n + n + 2) * (n^2 - 2*k^2 + 2*k*n + 5*n + 6)), k=0..n), n=0..infinity), 120);

RealDigits[NIntegrate[QPochhammer[x]^2, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

Equals Sum_{n>=0} Sum_{k=0..n} 8*(n+1)*(-1)^n / ((n^2 - 2*k^2 + 2*k*n + n + 2) * (n^2 - 2*k^2 + 2*k*n + 5*n + 6)).

Equals Sum_{n>=0} Sum_{j=-floor(n/2)..floor(n/2)} (-1)^(n+j) / (n*(n+1)/2 - j*(3*j-1)/2 + 1).

A258404 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 29 2015

			0.16182024229485656180261334985786534313068578288018990398...

Vaclav Kotesovec, The integration of q-series

Cf. A000727, A258232, A258406, A258407, A258404, A258405.

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

nmax=200; p=1; q4=Table[PrintTemporary[n]; p=Expand[p*(1-x^n)^4]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]],{n,1,nmax}]; q4n=N[q4,1000]; Table[SequenceLimit[Take[q4n,j]],{j,Length[q4n]-100,Length[q4n],10}]
NSum[2*(-1)^m*Pi/Cosh[Sqrt[7 - 4*m + 12*m^2]*Pi/2], {m, -Infinity, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100] (* Vaclav Kotesovec, Dec 04 2015 *)
RealDigits[NIntegrate[QPochhammer[x]^4, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

default(realprecision, 93);
b(n) = cosh(sqrt(7 - 4*n + 12*n^2)*Pi/2);
2*Pi*(1/b(0) + sumalt(n=1, (-1)^n*(1/b(n) + 1/b(-n)))) \\ Gheorghe Coserea, Sep 26 2018

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

More digits from Vaclav Kotesovec, Oct 10 2023

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

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A010816 Expansion of Product_{k>=1} (1 - x^k)^3.

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A258232 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

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A258406 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^2 dx.

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A258404 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.

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

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