A010815 - OEIS

%I #211 Oct 14 2024 12:27:23

%S 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,

%T 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,

%U 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

%N From Euler's Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).

%C When convolved with the partition numbers A000041 gives 1, 0, 0, 0, 0, ...

%C Also, number of different partitions of n into parts of -1 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004

%C The comment that "when convolved with the partition numbers gives [1, 0, 0, 0, ...]" is equivalent to row sums of triangle A145975 = [1, 0, 0, 0, ...]; where A145975 is a partition number convolution triangle. - _Gary W. Adamson_, Oct 25 2008

%C When convolved with n-th partial sums of A000041 = the binomial sequence starting (1, n, ...). Example: A010815 convolved with A014160 (partial sum operation applied thrice to the partition numbers) = (1, 3, 6, 10, ...). - _Gary W. Adamson_, Nov 11 2008

%C (A000012^(-n) * A000041) convolved with A010815 = n-th row of the inverse of Pascal's triangle, (as a vector, followed by zeros); where A000012^(-1) = the pairwise difference operator. Example: (A000012^(-4) * A000041) convolved with A010815 = (1, -4, 6, -4, 1, 0, 0, 0, ...). - _Gary W. Adamson_, Nov 11 2008

%C Also sum of [product of (1-2/(hook lengths)^2)] over all partitions of n. - _Wouter Meeussen_, Sep 16 2010

%C Cayley (1895) begins article 387 with "Write for shortness sqrt(2k'K / pi) / [1-q^{2m-1}]^2 = G, ..." which is a convoluted way of writing G = [1-q^{2m}] = (1-q^2)(1-q^4)... - _Michael Somos_, Aug 01 2011

%C This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^3, b = x. - _Michael Somos_, Jan 21 2012

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

%C Number 1 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - _Michael Somos_, May 04 2016

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 825.

%D 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.

%D T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.

%D A. Cayley, An Elementary Treatise on Elliptic Functions, G. Bell and Sons, London, 1895, p. 295, Art. 387.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5g].

%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.12) and (32.13).

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 353.

%D B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 70.

%D A. Weil, Number theory: an approach through history; from Hammurapi to Legendre, Birkhäuser, Boston, 1984; see p. 186.

%H Seiichi Manyama, <a href="/A010815/b010815.txt">Table of n, a(n) for n = 0..10000</a> (first 1002 terms from T. D. Noe)

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 825.

%H George E. Andrews, <a href="http://dx.doi.org/10.1090/S0273-0979-07-01180-9">Euler's "De Partitio Numerorum"</a>, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.

%H A. A. Bennett, <a href="http://www.jstor.org/stable/2300869">Problem 3553</a>, Amer. Math. Monthly, 39 (1932), 300.

%H M. Boylan, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00037-9">Exceptional congruences for the coefficients of certain eta-product newforms</a>, J. Number Theory 98 (2003), no. 2, 377-389.

%H D. Bump, <a href="https://doi.org/10.1017/CBO9780511609572">Automorphic Forms and Representations</a>, Cambr. Univ. Press, 1997, p. 29.

%H S. Cooper and M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00079-7">Results of Hurwitz type for three squares</a>, Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).

%H L. Euler, <a href="https://arxiv.org/abs/math/0411454">The expansion of the infinite product (1-x)(1-xx)(1-x^3)...</a>, arXiv:math/0411454 [math.HO], 2004.

%H L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E541.html">Evolutio producti infiniti (1-x)(1-xx)(1-x^3)...</a>

%H S. R. Finch, <a href="https://arxiv.org/abs/math/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.

%H Fern Gossow, <a href="https://arxiv.org/abs/2410.05678">Lyndon-like cyclic sieving and Gauss congruence</a>, arXiv:2410.05678 [math.CO], 2024. See p. 26.

%H H. Gupta, <a href="/A001482/a001482.pdf">On the coefficients of the powers of Dedekind's modular form</a> (annotated and scanned copy)

%H H. Gupta, <a href="https://doi.org/10.1112/jlms/s1-39.1.433">On the coefficients of the powers of Dedekind's modular form</a>, J. London Math. Soc., 39 (1964), 433-440.

%H K. Harada, <a href="http://dx.doi.org/10.4171/090">"Moonshine" of Finite Groups</a>, European Math. Soc., 2010, p. 17.

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Janjic2/janjic53.html">A Generating Function for Numbers of Insets</a>, Journal of Integer Sequences, 17, 2014, #14.9.7.

%H Vaclav Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>

%H S. C. Milne, <a href="http://dx.doi.org/10.1023/A:1014865816981">Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions</a>, Ramanujan J., 6 (2002), 7-149. (See (1.10).)

%H Tim Silverman, <a href="https://arxiv.org/abs/1612.08085">Counting Cliques in Finite Distant Graphs</a>, arXiv preprint arXiv:1612.08085 [math.CO], 2016.

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>

%H D. Zagier, <a href="http://dx.doi.org/10.1007/978-3-540-74119-0">Elliptic modular forms and their applications</a> in "The 1-2-3 of modular forms", Springer-Verlag, 2008.

%H Robert M. Ziff, <a href="http://dx.doi.org/10.1088/0305-4470/28/5/013">On Cardy's formula for the critical crossing probability in 2d percolation</a>, J. Phys. A. 28, 1249-1255 (1995).

%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>

%F 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.

%F Expansion of the Dedekind eta function without the q^(1/24) factor in powers of q.

%F Euler transform of period 1 sequence [ -1, -1, -1, ...].

%F G.f.: (q; q)_{infinity} = 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.

%F 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

%F a(n) = A067661(n) - A067659(n). - _Jon Perry_, Jun 17 2003

%F 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

%F G.f.: q^(-1/24) * eta(t), where q = exp(2 Pi i t) and eta is the Dedekind eta function.

%F G.f.: 1 - x - x^2(1-x) - x^3(1-x)(1-x^2) - ... - _Jon Perry_, Aug 07 2004

%F 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

%F 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

%F 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

%F 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

%F 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

%F 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

%F a(n) = -(1/n)*Sum_{k=1..n} sigma(k)*a(n-k). - _Vladeta Jovovic_, Aug 28 2002

%F 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

%F 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

%F G.f.: exp( Sum_{n>=1} -sigma(n)*x^n/n ). - _Seiichi Manyama_, Mar 04 2017

%F 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

%F 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

%e G.f. = 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 - x^35 - x^40 + ...

%e G.f. = q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + q^529 + q^625 + ...

%e From _Seiichi Manyama_, Mar 04 2017: (Start)

%e G.f.

%e = 1 + (-x - 3*x^2/2 - 4*x^3/3 -  7*x^4/4  -  6*x^5/5 - ...)

%e      + 1/2 * (x^2   + 3*x^3   + 59*x^4/12 + 15*x^5/2 + ...)

%e               + 1/6 * (-x^3   -  9*x^4/2  - 43*x^5/4 - ...)

%e                          + 1/24 * (x^4    +  6*x^5   + ...)

%e                                    + 1/120 * (-x^5   - ...)

%e                                              + ...

%e = 1 - x - x^2 + x^5 + .... (End)

%p A010815 := mul((1-x^m), m=1..100);

%p A010815 := proc(n) local x,m;

%p     product(1-x^m,m=1..n) ;

%p     expand(%) ;

%p     coeff(%,x,n) ;

%p end proc: # _R. J. Mathar_, Jun 18 2016

%p A010815 := proc(n) 24*n + 1; if issqr(%) then sqrt(%);

%p (-1)^irem(iquo(% + irem(%, 6), 6), 2) else 0 fi end: # _Peter Luschny_, Oct 02 2022

%t a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}], {x, 0, n}]; (* _Michael Somos_, Nov 15 2011 *)

%t 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 *)

%t CoefficientList[ Series[ Product[(1 - x^k), {k, 1, 70}], {x, 0, 70}], x]

%t (* hooklength[ ] cfr A047874 *) Table[ Tr[ ( Times@@(1-2/Flatten[hooklength[ # ]]^2) )&/@ Partitions[n] ],{n,26}] (* _Wouter Meeussen_, Sep 16 2010 *)

%t CoefficientList[ Series[ QPochhammer[q], {q, 0, 100}], q] (* _Jean-François Alcover_, Dec 04 2013 *)

%t a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ[m], KroneckerSymbol[ 12, m], 0]]; (* _Michael Somos_, Jun 04 2015 *)

%t 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 *)

%t 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 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n)), n))}; /* _Michael Somos_, Jun 05 2002 */

%o (PARI) {a(n) = polcoeff( prod( k=1, n, 1 - x^k, 1 + x * O(x^n)), n)}; /* _Michael Somos_, Nov 19 2011 */

%o (PARI) {a(n) = if( issquare( 24*n + 1, &n), kronecker( 12, n))}; /* _Michael Somos_, Feb 26 2006 */

%o (PARI) {a(n) = if( issquare( 24*n + 1, &n), if( (n%2) && (n%3), (-1)^round( n/6 )))}; /* _Michael Somos_, Feb 26 2006 */

%o (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 */

%o (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q))} \\ _Altug Alkan_, Mar 21 2018

%o (Magma) Coefficients(&*[1-x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // _Vincenzo Librandi_, Jan 15 2017

%o (Julia) # DedekindEta is defined in A000594.

%o A010815List(len) = DedekindEta(len, 1)

%o A010815List(93) |> println # _Peter Luschny_, Mar 09 2018

%o (Python)

%o from math import isqrt

%o def A010815(n):

%o     m = isqrt(24*n+1)

%o     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

%o (Julia)

%o function A010815(n)

%o     r = 24 * n + 1

%o     m = isqrt(r)

%o     m * m != r && return 0

%o     iseven(div(m + m % 6, 6)) ? 1 : -1

%o end # _Peter Luschny_, Sep 09 2021

%Y Cf. A000041, A001318 (characteristic function), A000326, A080995.

%Y Cf. A067659, A067661.

%Y Cf. A145975, A002865, A014160.

%Y Cf. also A170925, A143374, A194087, A242168, A258232, A143062, A203568.

%K sign,nice,easy

%O 0,1

%A _N. J. A. Sloane_

%E Additional comments from _Michael Somos_, Jun 05 2002