login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A010815 From Euler's Pentagonal Theorem: coefficient of q^n in Product (1-q^m), m=1.. infinity. 121
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; text; internal format)
OFFSET

0,1

COMMENTS

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

a(n)=A067661(n)-A067659(n) (number of partitions into an even number of distinct parts - number of partitions into an odd number of distinct parts). - Jon Perry, 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. - Gary W. Adamson, 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,...). - Gary W. Adamson, 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,...). - Gary W. Adamson, Nov 11 2008

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

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

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).

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".

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.

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].

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

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, Birkhaeuser, 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.

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).

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).

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.

M. Janjic, A Generating Function for Numbers of Insets, Journal of Integer Sequences, 17, 2014, #14.9.7.

V. 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).)

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

D. 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

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 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

A147843 = (-n) * a(n). - Gary W. Adamson, Nov 15 2008

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

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

EXAMPLE

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 + ...

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

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-Fran├žois Alcover, Dec 04 2013 *)

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

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) = 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 */

CROSSREFS

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

Cf. A067659, A067661.

Cf. A145975, A002865, A014160, A147843. - Gary W. Adamson, Nov 15 2008

Cf. also A170925, A143374, A194087, A242168, A258232.

Sequence in context: A133080 A133985 A143062 * A206958 A206959 A080995

Adjacent sequences:  A010812 A010813 A010814 * A010816 A010817 A010818

KEYWORD

sign,nice,easy,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Additional comments from Michael Somos, Jun 05 2002

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 26 07:45 EDT 2016. Contains 274210 sequences.