A010815 From Euler's Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).
1, -1, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0
Examples
G.f. = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 - x^35 - x^40 + ...
G.f. = q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + q^529 + q^625 + ...
From _Seiichi Manyama_, Mar 04 2017: (Start)
G.f.
= 1 + (-x - 3*x^2/2 - 4*x^3/3 - 7*x^4/4 - 6*x^5/5 - ...)
+ 1/2 * (x^2 + 3*x^3 + 59*x^4/12 + 15*x^5/2 + ...)
+ 1/6 * (-x^3 - 9*x^4/2 - 43*x^5/4 - ...)
+ 1/24 * (x^4 + 6*x^5 + ...)
+ 1/120 * (-x^5 - ...)
+ ...
= 1 - x - x^2 + x^5 + .... (End)
References
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 825.
- B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054. See page 3.
- T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.
- A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 295, Art. 387.
- L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5g].
- N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.12) and (32.13).
- G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 86.
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 353.
- B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 70.
- A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 186.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1002 terms from T. D. Noe)
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 825.
- George E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.
- A. A. Bennett, Problem 3553, Amer. Math. Monthly, 39 (1932), 300.
- M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
- D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
- S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares, Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).
- Leonhard Euler, The expansion of the infinite product (1-x)(1-xx)(1-x^3)..., arXiv:math/0411454 [math.HO], 2004.
- Leonhard Euler, Evolutio producti infiniti (1-x)(1-xx)(1-x^3)...
- S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
- Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
- H. Gupta, On the coefficients of the powers of Dedekind's modular form (annotated and scanned copy)
- H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.
- K. Harada, "Moonshine" of Finite Groups, European Math. Soc., 2010, p. 17.
- Milan Janjic, A Generating Function for Numbers of Insets, Journal of Integer Sequences, 17, 2014, #14.9.7.
- Vaclav Kotesovec, The integration of q-series
- S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149. (See (1.10).)
- Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Dedekind Eta Function
- Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
- Eric Weisstein's World of Mathematics, q-Pochhammer Symbol
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
- Eric Weisstein's World of Mathematics, Quintuple Product Identity
- Don Zagier, Elliptic modular forms and their applications in "The 1-2-3 of modular forms", Springer-Verlag, 2008.
- Robert M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A. 28, 1249-1255 (1995).
- Index entries for expansions of Product_{k >= 1} (1-x^k)^m
Crossrefs
Programs
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Julia
# DedekindEta is defined in A000594. A010815List(len) = DedekindEta(len, 1) A010815List(93) |> println # Peter Luschny, Mar 09 2018
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Julia
function A010815(n) r = 24 * n + 1 m = isqrt(r) m * m != r && return 0 iseven(div(m + m % 6, 6)) ? 1 : -1 end # Peter Luschny, Sep 09 2021
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Magma
Coefficients(&*[1-x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Vincenzo Librandi, Jan 15 2017
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Maple
A010815 := mul((1-x^m), m=1..100); A010815 := proc(n) local x,m; product(1-x^m,m=1..n) ; expand(%) ; coeff(%,x,n) ; end proc: # R. J. Mathar, Jun 18 2016 A010815 := proc(n) 24*n + 1; if issqr(%) then sqrt(%); (-1)^irem(iquo(% + irem(%, 6), 6), 2) else 0 fi end: # Peter Luschny, Oct 02 2022
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Mathematica
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Nov 15 2011 *) a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^(3/2)], {x, 0, n + Floor@Sqrt[n]}] // Normal // TrigToExp) /. {y -> -x^(1/2)}, {x, 0, n}]]; (* Michael Somos, Nov 15 2011 *) CoefficientList[ Series[ Product[(1 - x^k), {k, 1, 70}], {x, 0, 70}], x] (* hooklength[ ] cfr A047874 *) Table[ Tr[ ( Times@@(1-2/Flatten[hooklength[ # ]]^2) )&/@ Partitions[n] ],{n,26}] (* Wouter Meeussen, Sep 16 2010 *) CoefficientList[ Series[ QPochhammer[q], {q, 0, 100}], q] (* Jean-François Alcover, Dec 04 2013 *) a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ[m], KroneckerSymbol[ 12, m], 0]]; (* Michael Somos, Jun 04 2015 *) nmax = 100; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = -1; Do[Do[poly[[j + 1]] -= poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly (* Vaclav Kotesovec, May 04 2018 *) Table[m = (1 + Sqrt[1 + 24*k])/6; If[IntegerQ[m], (-1)^m, 0] + If[IntegerQ[m - 1/3], (-1)^(m - 1/3), 0], {k, 0, 100}] (* Vaclav Kotesovec, Jul 09 2020 *) -
PARI
{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n)), n))}; /* Michael Somos, Jun 05 2002 */ -
PARI
{a(n) = polcoeff( prod( k=1, n, 1 - x^k, 1 + x * O(x^n)), n)}; /* Michael Somos, Nov 19 2011 */ -
PARI
{a(n) = if( issquare( 24*n + 1, &n), kronecker( 12, n))}; /* Michael Somos, Feb 26 2006 */ -
PARI
{a(n) = if( issquare( 24*n + 1, &n), if( (n%2) && (n%3), (-1)^round( n/6 )))}; /* Michael Somos, Feb 26 2006 */ -
PARI
{a(n) = my(A); if( n<0, 0, A = 1 + O(x^n); polcoeff( sum( k=1, (sqrtint( 8*n + 1)-1) \ 2, A *= x^k / (x^k - 1) + x * O(x^(n - (k^2-k)/2)), 1), n))}; /* Michael Somos, Aug 18 2006 */ -
PARI
lista(nn) = {q='q+O('q^nn); Vec(eta(q))} \\ Altug Alkan, Mar 21 2018 -
Python
from math import isqrt def A010815(n): m = isqrt(24*n+1) return 0 if m**2 != 24*n+1 else ((-1)**((m-1)//6) if m % 6 == 1 else (-1)**((m+1)//6)) # Chai Wah Wu, Sep 08 2021
Formula
a(n) = (-1)^m if n is of the form m(3m+-1)/2; otherwise a(n)=0. The values of n such that |a(n)|=1 are the generalized pentagonal numbers, A001318. The values of n such that a(n)=0 is A090864.
Expansion of the Dedekind eta function without the q^(1/24) factor in powers of q.
Euler transform of period 1 sequence [ -1, -1, -1, ...].
G.f.: (q; q){oo} = Product{k >= 1} (1-q^k) = Sum_{n=-oo..oo} (-1)^n*q^(n*(3n+1)/2). The first notation is a q-Pochhammer symbol.
Expansion of f(-x) := f(-x, -x^2) in powers of x. A special case of Ramanujan's general theta function; see Berndt reference. - Michael Somos, Apr 08 2003
Expansion of f(x^5, x^7) - x * f(x, x^11) in powers of x where f(, ) is Ramanujan's general theta function. - Michael Somos, Jan 21 2012
G.f.: q^(-1/24) * eta(t), where q = exp(2 Pi i t) and eta is the Dedekind eta function.
G.f.: 1 - x - x^2(1-x) - x^3(1-x)(1-x^2) - ... - Jon Perry, Aug 07 2004
Given g.f. A(x), then B(q) = q * A(q^3)^8 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - v^3 + 16*u*w^2. - Michael Somos, May 02 2005
Given g.f. A(x), then B(q) = q * A(q^24) satisfies 0 = f(B(q), B(x^q), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1^9*u3*u6^3 - u2^9*u3^4 + 9*u1^4*u2*u6^8. - Michael Somos, May 02 2005
a(n) = b(24*n + 1) where b() is multiplicative with b(p^2e) = (-1)^e if p == 5 or 7 (mod 12), b(p^2e) = +1 if p == 1 or 11 (mod 12) and b(p^(2e-1)) = b(2^e) = b(3^e) = 0 if e>0. - Michael Somos, May 08 2005
Given g.f. A(x), then B(q) = q * A(q^24) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^16*w^8 - v^24 + 16*u^8*w^16. - Michael Somos, May 08 2005
a(n) = (-1)^n * A121373(n). a(25*n + 1) = -a(n). a(5*n + 3) = a(5*n + 4) = 0. a(5*n) = A113681(n). a(5*n + 2) = - A116915(n). - Michael Somos, Feb 26 2006
G.f.: 1 + Sum_{k>0} (-1)^k * x^((k^2 + k) / 2) / ((1 - x) * (1 - x^2) * ... * (1 - x^k)). - Michael Somos, Aug 18 2006
a(n) = -(1/n)*Sum_{k=1..n} sigma(k)*a(n-k). - Vladeta Jovovic, Aug 28 2002
G.f.: A(x) = 1 - x/G(0); G(k) = 1 + x - x^(k+1) - x*(1-x^(k+1))/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 25 2012
Expansion of f(-x^2) * chi(-x) = psi(-x) * chi(-x^2) = psi(x) * chi(-x)^2 = f(-x^2)^2 / psi(x) = phi(-x) / chi(-x) = phi(-x^2) / chi(x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Nov 16 2015
G.f.: exp( Sum_{n>=1} -sigma(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017
G.f.: Sum_{n >= 0} x^(n*(2*n-1))*(2*x^(2*n) - 1)/Product_{k = 1..2*n} 1 - x^k. - Peter Bala, Feb 02 2021
The g.f. A(x) satisfies A(x^2) = Sum_{n >= 0} x^(n*(n+1)/2) * Product_{k >= n+1} 1 - x^k = 1 - x^2 - x^4 + x^10 + x^14 - x^24 - x^30 + + - - .... - Peter Bala, Feb 12 2021
For m >= 0, A(x) = (1 - x)*(1 - x^2)*...*(1 - x^m) * Sum_{n >= 0} (-1)^n * x^(n*(n+2*m+1)/2) /(Product_{k = 1..n} 1 - x^k). - Peter Bala, Feb 03 2025
From Friedjof Tellkamp, Mar 19 2025: (Start)
Sum_{n>=1} a(n)/n = 6 - 4*Pi/sqrt(3).
Sum_{n>=1} a(n)/n^2 = -108 + 16*sqrt(3)*Pi + 2*Pi^2.
Sum_{n>=1} a(n)/n^k = Sum_{i=0..k} 6^(k-i)*C(-k, k-i)*A(i), where A(i)=(2^i-2)*(3^i-3)*zeta(i) for even i, and A(i)=-G(i/2-1/2)*(2^i+2)*(2*Pi)^i/(sqrt(3)*Gamma(i+1)) for odd i, with G(n>0) as the Glaisher's numbers (A002111) and G(0)=1/2. (End)
Extensions
Additional comments from Michael Somos, Jun 05 2002
Comments