A145975 -id:A145975 - cp's OEIS frontend

A010815 From Euler's Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).

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

Offset: 0

N. J. A. Sloane

When convolved with the partition numbers A000041 gives 1, 0, 0, 0, 0, ...
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) * A000041) 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". - Michael Somos, May 04 2016

			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 + ...
From _Seiichi Manyama_, Mar 04 2017: (Start)
G.f.
= 1 + (-x - 3*x^2/2 - 4*x^3/3 -  7*x^4/4  -  6*x^5/5 - ...)
     + 1/2 * (x^2   + 3*x^3   + 59*x^4/12 + 15*x^5/2 + ...)
              + 1/6 * (-x^3   -  9*x^4/2  - 43*x^5/4 - ...)
                         + 1/24 * (x^4    +  6*x^5   + ...)
                                   + 1/120 * (-x^5   - ...)
                                             + ...
= 1 - x - x^2 + x^5 + .... (End)

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.
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, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 86.
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, Birkhäuser, Boston, 1984; see p. 186.

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.
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
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).
Leonhard Euler, The expansion of the infinite product (1-x)(1-xx)(1-x^3)..., arXiv:math/0411454 [math.HO], 2004.
Leonhard Euler, Evolutio producti infiniti (1-x)(1-xx)(1-x^3)...
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
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.
Milan Janjic, A Generating Function for Numbers of Insets, Journal of Integer Sequences, 17, 2014, #14.9.7.
Vaclav 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).)
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
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
Don 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

Cf. A000041, A001318 (characteristic function), A000326, A080995.
Cf. A067659, A067661.
Cf. A145975, A002865, A014160.
Cf. also A170925, A143374, A194087, A242168, A258232, A143062, A203568, A002111, A380038.

# DedekindEta is defined in A000594.
A010815List(len) = DedekindEta(len, 1)
A010815List(93) |> println # Peter Luschny, Mar 09 2018

function A010815(n)
    r = 24 * n + 1
    m = isqrt(r)
    m * m != r && return 0
    iseven(div(m + m % 6, 6)) ? 1 : -1
end # Peter Luschny, Sep 09 2021

Coefficients(&*[1-x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Vincenzo Librandi, Jan 15 2017

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
A010815 := proc(n) 24*n + 1; if issqr(%) then sqrt(%);
(-1)^irem(iquo(% + irem(%, 6), 6), 2) else 0 fi end: # Peter Luschny, Oct 02 2022

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-François Alcover, Dec 04 2013 *)
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ[m], KroneckerSymbol[ 12, m], 0]]; (* Michael Somos, Jun 04 2015 *)
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 *)
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 *)

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n)), n))}; /* Michael Somos, Jun 05 2002 */

{a(n) = polcoeff( prod( k=1, n, 1 - x^k, 1 + x * O(x^n)), n)}; /* Michael Somos, Nov 19 2011 */

{a(n) = if( issquare( 24*n + 1, &n), kronecker( 12, n))}; /* Michael Somos, Feb 26 2006 */

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

{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 */

lista(nn) = {q='q+O('q^nn); Vec(eta(q))} \\ Altug Alkan, Mar 21 2018

from math import isqrt
def A010815(n):
    m = isqrt(24*n+1)
    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

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){oo} = 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.
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
a(n) = A067661(n) - A067659(n). - Jon Perry, Jun 17 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
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
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
G.f.: exp( Sum_{n>=1} -sigma(n)*x^n/n ). - Seiichi Manyama, Mar 04 2017
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
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
For m >= 0, A(x) = (1 - x)*(1 - x^2)*...*(1 - x^m) * Sum_{n >= 0} (-1)^n * x^(n*(n+2*m+1)/2) /(Product_{k = 1..n} 1 - x^k). - Peter Bala, Feb 03 2025
From Friedjof Tellkamp, Mar 19 2025: (Start)
Sum_{n>=1} a(n)/n = 6 - 4*Pi/sqrt(3).
Sum_{n>=1} a(n)/n^2 = -108 + 16*sqrt(3)*Pi + 2*Pi^2.
Sum_{n>=1} a(n)/n^k = Sum_{i=0..k} 6^(k-i)*C(-k, k-i)*A(i), where A(i)=(2^i-2)*(3^i-3)*zeta(i) for even i, and A(i)=-G(i/2-1/2)*(2^i+2)*(2*Pi)^i/(sqrt(3)*Gamma(i+1)) for odd i, with G(n>0) as the Glaisher's numbers (A002111) and G(0)=1/2. (End)

Additional comments from Michael Somos, Jun 05 2002

A027293 Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 3, 2, 1, 1, 7, 5, 3, 2, 1, 1, 11, 7, 5, 3, 2, 1, 1, 15, 11, 7, 5, 3, 2, 1, 1, 22, 15, 11, 7, 5, 3, 2, 1, 1, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 56, 42, 30, 22, 15, 11, 7, 5, 3, 2, 1, 1, 77

Offset: 1

Clark Kimberling

Triangle read by rows in which row n lists the first n partition numbers A000041 in decreasing order. - Omar E. Pol, Aug 06 2011
A027293 * an infinite lower triangular matrix with A010815 (1, -1, -1, 0, 0, 1, ...) as the main diagonal the rest zeros = triangle A145975 having row sums = [1, 0, 0, 0, ...]. These matrix operations are equivalent to the comment in A010815 stating "when convolved with the partition numbers = [1, 0, 0, 0, ...]. - Gary W. Adamson, Oct 25 2008
From Gary W. Adamson, Oct 26 2008: (Start)
Row sums = A000070: (1, 2, 4, 7, 12, 19, 30, 45, 67, ...);
(this triangle)^2 = triangle A146023. (End)
(1) It appears that P(n,k) is also the total number of occurrences of k in the last k sections of the set of partitions of n (cf. A182703). (2) It appears that P(n,k) is also the difference, between n and n-k, of the total number of occurrences of k in all their partitions (cf. A066633). - Omar E. Pol, Feb 07 2012
Sequence B is called a reverse reluctant sequence of sequence A, if B is a triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. The present sequence is the reverse reluctant sequence of (A000041(k-1)){k>=0}. - _Boris Putievskiy, Dec 14 2012

			The triangle P begins (with offsets 0 it is Pa):
n \ k  1  2  3  4  5  6  7  8  9 10 ...
1:     1
2:     1  1
3:     2  1  1
4:     3  2  1  1
5:     5  3  2  1  1
6:     7  5  3  2  1  1
7:    11  7  5  3  2  1  1
8:    15 11  7  5  3  2  1  1
9:    22 15 11  7  5  3  2  1  1
10:   30 22 15 11  7  5  3  2  1  1
... reformatted by _Wolfdieter Lang_, Apr 14 2021

Robert Price, Table of n, a(n) for n = 1..5050
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Every column of P is A000041.
Cf. A145975, A010815.
Cf. A000070, A146023.
Cf. A182700, A027293, A067687, A155002.
Cf. A343234 (L-eigen-matrix).

f[n_] := Block[{t = Flatten[Union /@ IntegerPartitions@n]}, Table[Count[t, i], {i, n}]]; Array[f, 13] // Flatten
t[n_, k_] := PartitionsP[n-k]; Table[t[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 24 2014 *)

P(n,k) = p(n-k) = A000041(n-k), n>=1, k>=1. - Omar E. Pol, Feb 15 2013
a(n) = A000041(m), where m = (t*t + 3*t + 4)/2 - n, t = floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 14 2012
From Wolfdieter Lang, Apr 14 2021: (Start)
Pa(n, m) = P(n+1, m+1) = A000041(n-m), for n >= m >= 0, and 0 otherwise, gives the Riordan matrix Pa = (P(x), x), of Toeplitz type, with the o.g.f. P(x) of A000041. The o.g.f. of triangle Pa (the o.g.f. of the row polynomials RPa(n, x) = Sum_{m=0..n} Pa(n, m)*x^m) is G(z, x) = P(z)/(1 - x*z).
The (infinite) matrix Pa has the 'L-eigen-sequence' B = A067687, that is, Pa*vec(B) = L*vec(B), with the matrix L with elements L(i, j) = delta(i, j-1) (Kronecker's delta symbol). For such L-eigen-sequences see the Bernstein and Sloane links under A155002.
Thanks to Gary W. Adamson for motivating me to look at such matrices and sequences. (End)

cp's OEIS Frontend

A146014 Triangle read by rows, square of A145975.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A010815 From Euler's Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Julia

Julia

Magma

Maple

Mathematica

PARI

PARI

PARI

PARI

PARI

PARI

Python

Formula

Extensions

A027293 Triangular array given by rows: P(n,k) is the number of partitions of n that contain k as a part.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula