%I A010815
%S A010815 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,
%T A010815 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,
%U A010815 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
%V A010815 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,
%W A010815 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,
%X A010815 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
%N A010815 From Euler's Pentagonal Theorem: coefficient of q^n in Product (1-q^m), 
               m=1.. infinity. Also the q-expansion of the Dedekind eta function 
               without the q^(1/24) factor.
%C A010815 When convolved with the partition numbers A000041 gives 1, 0, 0, 0, 0, 
               ...
%C A010815 Euler transform of period 1 sequence [ -1,-1,-1,...].
%C A010815 a(n)=A067659(n)-A067661(n) (number of partitions into an odd number of 
               distinct parts - number of partitions into an even number of distinct 
               parts) - Jon Perry (perry(AT)globalnet.co.uk), Jun 17 2003
%C A010815 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 A010815 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. 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2008]
%C A010815 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,...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 11 2008]
%C A010815 (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) * A000041j) 
               convolved with A010815 = (1, -4, 6, -4, 1, 0, 0, 0,...). [From Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Nov 11 2008]
%C A010815 A147843 convolved with A000041 = A000203 starting (0, 1, 3, 4, 7, 6,...), 
               where A147843 = (-n) * A010815(n). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Nov 15 2008]
%D A010815 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 A010815 G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 
               44 (No. 4, 2007), 561-573.
%D A010815 A. A. Bennett, Problem 3553, Amer. Math. Monthly, 39 (1932), 300.
%D A010815 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 A010815 M. Boylan, Exceptional congruences for the coefficients of certain eta-product 
               newforms, J. Number Theory 98 (2003), no. 2, 377-389.
%D A010815 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 
               2nd. ed. 1949, p. 116, Problem 18.
%D A010815 D. Bump, Automorphic Forms..., Cambridge Univ. Press, p. 1997 p. 29.
%D A010815 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 104, [5g].
%D A010815 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 77, Eq. (32.12) and (32.13).
%D A010815 H. Gupta, On the coefficients of the powers of Dedekind's modular form, 
               J. London Math. Soc., 39 (1964), 433-440.
%D A010815 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1979, Theorem 353.
%D A010815 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).)
%D A010815 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, 
               p. 70.
%D A010815 A. Weil, Number theory: an approach through history; from Hammurapi to 
               Legendre, Birkhaeuser, Boston, 1984; see p. 186.
%D A010815 Robert M. Ziff, "On Cardy's formula for the critical crossing probability 
               in 2d percolation," J. Phys. A. 28, 1249-1255 (1995).
%H A010815 T. D. Noe, <a href="b010815.txt">Table of n, a(n) for n=0..1001</a>
%H A010815 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A010815 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 A010815 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411454">The expansion 
               of the infinite product (1-x)(1-xx)(1-x^3)...</a>
%H A010815 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E541.html">
               Evolutio producti infiniti (1-x)(1-xx)(1-x^3)...</a>
%H A010815 S. R. Finch, <a href="http://arXiv.org/abs/math.NT/0701251">Powers of 
               Euler's q-Series</a>, (arXiv:math.NT/0701251).
%H A010815 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DedekindEtaFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A010815 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RamanujanThetaFunctions.html">Link to a section of The World of Mathematics.</
               a>
%H A010815 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               q-PochhammerSymbol.html">Link to a section of The World of Mathematics.</
               a>
%H A010815 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A010815 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>
%H A010815 <a href="Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k 
               >= 1} (1-x^k)^m</a>
%F A010815 a(n) = (-1)^m if n is of the form m(3m+-1)/2; otherwise a(n)=0. These 
               values of n are the pentagonal numbers, A000326.
%F A010815 G.f.: (q; q)_{infinity} = product_{k >= 1} (1-q^k) = sum_{n=-infinity..infinity} 
               (-1)^n*q^(n*(3n+1)/2). The first notation is a q-Pochhamer symbol.
%F A010815 G.f.: f(-q) = f(-q, -q^2), a special case of Ramanujan's theta function; 
               see Berndt reference. - Michael Somos, Apr 08 2003
%F A010815 G.f.: q^(-1/24)*eta(z), where q=exp(2 Pi i z) and eta is the Dedekind 
               eta function.
%F A010815 G.f.: 1 - x - x^2(1-x) - x^3(1-x)(1-x^2) - ... - Jon Perry (perry(AT)globalnet.co.uk), 
               Aug 07 2004
%F A010815 Given g.f. A(x), then B(x)=x*A(x^3)^8 satisfies 0=f(B(x), B(x^2), B(x^4)) 
               where f(u, v, w)=u^2*w -v^3 +16*u*w^2. - Michael Somos May 02 2005
%F A010815 Given g.f. A(x), then B(x)=x*A(x^24) satisfies 0=f(B(x), B(x^2), B(x^3), 
               B(x^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 A010815 a(n)=b(24n+1) where b(n) is multiplicative and 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 A010815 Given g.f. A(x), then B(x)=x*A(x^24) satisfies 0=f(B(x), B(x^2), B(x^4)) 
               where f(u, v, w)=u^16*w^8-v^24+16*u^8*w^16. - Michael Somos May 08 
               2005
%F A010815 a(25n+1)=-a(n). a(5n+3)=a(5n+4)=0. a(5n)=A113681(n). a(5n+2)=-A116915(n). 
               - Michael Somos Feb 26 2006
%F A010815 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 A010815 a(n) = -(1/n)*Sum_{k=1..n} sigma(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Aug 28 2002
%e A010815 eta(24z)=q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + q^529 +...
%p A010815 A010815 := mul((1-x^m), m=1..100);
%t A010815 CoefficientList[ Series[ Product[(1 - x^k), {k, 1, 70}], {x, 0, 70}], 
               x]
%o A010815 (PARI) a(n)=if(n<0,0,polcoeff(eta(x+x*O(x^n)),n))
%o A010815 (PARI) {a(n)=if(issquare(24*n+1, &n), kronecker(12, n))} /* Michael Somos 
               Feb 26 2006 */
%o A010815 (PARI) {a(n)=if(issquare(24*n+1, &n), if((n%2)&(n%3), (-1)^round(n/6)))} 
               /* Michael Somos Feb 26 2006 */
%o A010815 (PARI) {a(n)=local(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 */
%Y A010815 Cf. A000041, A001318, A000326. A080995(n)=|a(n)|.
%Y A010815 Cf. A067659, A067661.
%Y A010815 A145975 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 25 2008]
%Y A010815 Cf. A002865, A014160 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 
               11 2008]
%Y A010815 A147843 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]
%Y A010815 Sequence in context: A115512 A115513 A133080 this_sequence A080995 A121373 
               A133985
%Y A010815 Adjacent sequences: A010812 A010813 A010814 this_sequence A010816 A010817 
               A010818
%K A010815 sign,nice,easy
%O A010815 0,1
%A A010815 N. J. A. Sloane (njas(AT)research.att.com).
%E A010815 Additional comments from Michael Somos, Jun 05 2002