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A010815
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From Euler's Pentagonal Theorem: coefficient of q^n in Product (1-q^m), m=1.. infinity.
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96
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| When convolved with the partition numbers A000041 gives 1, 0, 0, 0, 0, ...
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
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]
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]
(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]
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]
Also sum of [product of (1-2/(hook lengths)^2)] over all partitions of n. [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 16 2010]
Cayley 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
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
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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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.
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.
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.
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].
Cooper, S. and Hirschhorn, M. D., Results of Hurwitz type for three squares. Discrete Math. 274 (2004), no. 1-3, 9-24. See P(q).
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.
K. Harada, "Moonshine" of Finite Groups, European Math. Soc., 2010, p. 17.
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.
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LINKS
| 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, 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.
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
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, 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
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
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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)_{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.
Expansion of f(-x) = f(-x, -x^2) in powers of x. A special case of Ramanujan's 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 two variable theta function. - Michael Somos, Jan 21 2012
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(24*n + 1) where b(n) 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(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(n)| = A080995(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 (vladeta(AT)eunet.rs), 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 Euler's kind, 1-step ). - Sergei N. Gladkovskii, Jan 25 2012
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EXAMPLE
| 1 - x - x^2 + x^5 + x^7 - x^12 - x^15 + x^22 + x^26 - x^35 - x^40 + ...
q - q^25 - q^49 + q^121 + q^169 - q^289 - q^361 + q^529 + q^625 + ...
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MAPLE
| A010815 := mul((1-x^m), m=1..100);
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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}] [From Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 16 2010]
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PROG
| (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) = 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 */
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CROSSREFS
| Cf. A000041, A001318 (characteristic function), A000326, A080995.
Cf. A067659, A067661.
Cf. A145975, A002865, A014160, A147843 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]
Cf. also A170925, A143374, A194087.
Sequence in context: A115512 A115513 A133080 * A080995 A121373 A133985
Adjacent sequences: A010812 A010813 A010814 * A010816 A010817 A010818
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KEYWORD
| sign,nice,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Additional comments from Michael Somos, Jun 05 2002
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