0,1

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

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

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

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

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

G. 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.

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.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.

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

D. Bump, Automorphic Forms..., Cambridge Univ. Press, p. 1997 p. 29.

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).

H. Gupta, On the coefficients of the powers of Dedekind's modular form, J. London Math. Soc., 39 (1964), 433-440.

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

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).)

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

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

Robert M. Ziff, "On Cardy's formula for the critical crossing probability in 2d percolation," J. Phys. A. 28, 1249-1255 (1995).

T. D. Noe, Table of n, a(n) for n=0..1001

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

L. Euler, The expansion of the infinite product (1-x)(1-xx)(1-x^3)...

L. Euler, Evolutio producti infiniti (1-x)(1-xx)(1-x^3)...

S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Eric Weisstein's World of Mathematics, Pentagonal Number Theorem

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

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.

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.

G.f.: f(-q) = f(-q, -q^2), a special case of Ramanujan's theta function; see Berndt reference. - Michael Somos, Apr 08 2003

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

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

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

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

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

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

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

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 (vladeta(AT)Eunet.yu), Aug 28 2002

eta(24z)=q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + q^529 +...

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

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

(PARI) a(n)=if(n<0, 0, polcoeff(eta(x+x*O(x^n)), n))

(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)=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 */

Cf. A000041, A001318, A000326. A080995(n)=|a(n)|.

Cf. A067659, A067661.

Sequence in context: A115512 A115513 A133080 this_sequence A080995 A121373 A133985

Adjacent sequences: A010812 A010813 A010814 this_sequence A010816 A010817 A010818

sign,nice,easy

njas

Additional comments from Michael Somos, Jun 05 2002