A106459 -id:A106459 - cp's OEIS frontend

A010054 a(n) = 1 if n is a triangular number, otherwise 0.

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

This is essentially the q-expansion of the Jacobi theta function theta_2(q). (In theta_2 one has to ignore the initial factor of 2*q^(1/4) and then replace q by q^(1/2). See also A005369.) - N. J. A. Sloane, Aug 03 2014
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's theta function f(a, b) = Sum_{n=-inf..inf} a^(n*(n+1)/2) * b^(n*(n-1)/2).
This sequence is the concatenation of the base-b digits in the sequence b^n, for any base b >= 2. - Davis Herring (herring(AT)lanl.gov), Nov 16 2004
Number of partitions of n into distinct parts such that the greatest part equals the number of all parts, see also A047993; a(n)=A117195(n,0) for n > 0; a(n) = 1-A117195(n,1) for n > 1. - Reinhard Zumkeller, Mar 03 2006
Triangle T(n,k), 0 <= k <= n, read by rows, given by A000007 DELTA A000004 where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
Convolved with A000041 = A022567, the convolution square of A000009. - Gary W. Adamson, Jun 11 2009
A008441(n) = Sum_{k=0..n} a(k)*a(n-k). - Reinhard Zumkeller, Nov 03 2009
Polcoeff inverse with alternate signs = A006950: (1, 1, 1, 2, 3, 4, 5, 7, ...). - Gary W. Adamson, Mar 15 2010
This sequence is related to Ramanujan's two-variable theta functions because this sequence is also the characteristic function of generalized hexagonal numbers. - Omar E. Pol, Jun 08 2012
Number 3 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
Number of partitions of n into consecutive parts that contain 1 as a part, n >= 1. - Omar E. Pol, Nov 27 2020
The constant whose decimal expansion is this sequence is irrational (Mahler, 1981). The constant whose expansion in any base b >= 2 is this sequence is irrational (Bundschuh, 1984). - Amiram Eldar, Mar 23 2025

			G.f. = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 + x^45 + x^55 + x^66 + ...
G.f. for B(q) = q * A(q^8): q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + q^225 + q^289 + q^361 + ...
From _Philippe Deléham_, Jan 04 2008: (Start)
As a triangle this begins:
  1;
  1, 0;
  1, 0, 0;
  1, 0, 0, 0;
  1, 0, 0, 0, 0;
  1, 0, 0, 0, 0, 0;
  ...  (End)

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1999, p. 103.
Michael D. Hirschhorn, The Power of q, Springer, 2017. See Psi, page 9.
Jules Tannery and Jules Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
Edmund T. Whittaker and George N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum, Vol. 17, No. 3 (2022), 129-141. See Conjecture 4.4, p. 137.
Peter Bundschuh, Generalization of a recent irrationality result of Mahler, Journal of Number Theory, Vol. 19, No. 2 (1984), pp. 248-253.
S. Cooper and M. D. Hirschhorn, Results of Hurwitz type for three squares, Discrete Math. 274 (2004), no. 1-3, 9-24. See psi(q).
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See p. 2.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 2.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Michael D. Hirschhorn and James A. Sellers, A Congruence Modulo 3 for Partitions into Distinct Non-Multiples of Four, Article 14.9.6, Journal of Integer Sequences, Vol. 17 (2014).
Kurt Mahler, On some irrational decimal fractions, Journal of Number Theory, Vol. 13, No. 2 (1981), pp. 268-269.
Ken Ono, Sinai Robins, and Patrick T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, Volume 50, Issue 1-2 (August 1995), pp. 73-94, Proposition 1; ResearchGate link; author's copy.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Michael Somos, A Multisection of q-Series, 2017.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Wolfram Challenges, Separate Ones by Zeroes.
Index entries for characteristic functions.

Cf. A000217, A002448, A005369, A023531, A035214, A022567, A052343, A006950, A106459, A127648.
Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
Cf. A106507 (reciprocal series).

(def A010054 (mapcat #(cons 1 (replicate % 0)) (range))) ; Tony Zorman, Apr 03 2023

a010054 = a010052 . (+ 1) . (* 8)
a010054_list = concatMap (\x -> 1 : replicate x 0) [0..]
-- Reinhard Zumkeller, Feb 12 2012, Oct 22 2011, Apr 02 2011

Basis( ModularForms( Gamma0(16), 1/2), 362) [2] ; /* Michael Somos, Jun 10 2014 */

A010054 := proc(n)
    if issqr(1+8*n) then
        1;
    else
        0;
    end if;
end proc:
seq(A010054(n),n=0..80) ; # R. J. Mathar, Feb 22 2021

a[ n_] := SquaresR[ 1, 8 n + 1] / 2; (* Michael Somos, Nov 15 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ (Series[ EllipticTheta[ 3, Log[y] / (2 I), x^2], {x, 0, n + Floor @ Sqrt[n]}] // Normal // TrigToExp) /. {y -> x}, {x, 0, n}]]; (* Michael Somos, Nov 15 2011 *)
Table[If[IntegerQ[(Sqrt[8n+1]-1)/2],1,0],{n,0,110}] (* Harvey P. Dale, Oct 29 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]; (* Michael Somos, Jul 01 2014 *)
Module[{tr=Accumulate[Range[20]]},If[MemberQ[tr,#],1,0]&/@Range[Max[tr]]] (* Harvey P. Dale, Mar 13 2023 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(x + A), n))}; /* Michael Somos, Mar 14 2011 */

{a(n) = issquare( 8*n + 1)}; /* Michael Somos, Apr 27 2000 */

a(n) = ispolygonal(n, 3); \\ Michel Marcus, Jan 22 2015

from sympy import integer_nthroot
def A010054(n): return int(integer_nthroot((n<<3)+1,2)[1]) # Chai Wah Wu, Nov 15 2022

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(-1, 0)
a = EulerTransform(b)
print([a(n) for n in range(88)]) # Peter Luschny, Nov 17 2022

Expansion of f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.
Expansion of q^(-1) * (phi(q) - phi(q^4)) / 2 in powers of q^8. - Michael Somos, Jul 01 2014
Expansion of q^(-1/8) * eta(q^2)^2 / eta(q) in powers of q. - Michael Somos, Apr 13 2005
Euler transform of period 2 sequence [ 1, -1, ...]. - Michael Somos, Mar 24 2003
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u6^3 + u2*u3^3 - u1*u2^2*u6. - Michael Somos, Apr 13 2005
a(n) = b(8*n + 1) where b()=A098108() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p > 2. - Michael Somos, Jun 06 2005
a(n) = A005369(2*n). - Michael Somos, Apr 29 2003
G.f.: theta_2(sqrt(q)) / (2 * q^(1/8)).
G.f.: 1 / (1 - x / (1 + x / (1 + x^1 / (1 - x / (1 + x / (1 + x^2 / (1 - x / (1 + x / (1 + x^3 / ...))))))))). - Michael Somos, May 11 2012
G.f.: Product_{k>0} (1-x^(2*k))/(1-x^(2*k-1)). - Vladeta Jovovic, May 02 2002
a(0)=1; for n>0, a(n) = A002024(n+1)-A002024(n). - Benoit Cloitre, Jan 05 2004
G.f.: Sum_{j>=0} Product_{k=0..j} x^j. - Jon Perry, Mar 30 2004
a(n) = floor((1-cos(Pi*sqrt(8*n+1)))/2). - Carl R. White, Mar 18 2006
a(n) = round(sqrt(2n+1)) - round(sqrt(2n)). - Hieronymus Fischer, Aug 06 2007
a(n) = ceiling(2*sqrt(2n+1)) - floor(2*sqrt(2n)) - 1. - Hieronymus Fischer, Aug 06 2007
a(n) = f(n,0) with f(x,y) = if x > 0 then f(x-y,y+1), otherwise 0^(-x). - Reinhard Zumkeller, Sep 27 2008
a(n) = A035214(n) - 1.
From Mikael Aaltonen, Jan 22 2015: (Start)
Since the characteristic function of s-gonal numbers is given by floor(sqrt(2n/(s-2) + ((s-4)/(2s-4))^2) + (s-4)/(2s-4)) - floor(sqrt(2(n-1)/(s-2) + ((s-4)/(2s-4))^2) + (s-4)/(2s-4)), by setting s = 3 we get the following: For n > 0, a(n) = floor(sqrt(2*n+1/4)-1/2) - floor(sqrt(2*(n-1)+1/4)-1/2).
(End)
a(n) = (-1)^n * A106459(n). - Michael Somos, May 04 2016
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(-1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A002448. - Michael Somos, May 05 2016
G.f.: Sum_{n >= 0} x^(n*(n+1)/2) = Product_{n >= 1} (1 - x^n)*(1 + x^n)^2 = Product_{n >= 1} (1 - x^(2*n))*(1 + x^n) = Product_{n >= 1} (1 - x^(2*n))/(1 - x^(2*n-1)). From the sum and product representations of theta_2(0, sqrt(q))/(2*q^(1/8)) function. The last product, given by Vladeta Jovovic above, is obtained from the second to last one by an Euler identity, proved via f(x) := Product_{n >= 1} (1 - x^(2*n-1))*Product_{n >= 1} (1 + x^n) = f(x^2), by moving the odd-indexed factors of the second product to the first product. This leads to f(x) = f(0) = 1. - Wolfdieter Lang, Jul 05 2016
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 08 2017
G.f.: Sum_{n >= 0} x^n * Product_{k >= n+1} (1 - x^(2*k)) = 1/(1 - x) * Sum_{n >= 0} x^(3*n) * Product_{k >= n+1} (1 - x^(2*k)) = 1/((1 - x)*(1 - x^3)) * Sum_{n >= 0} x^(5*n) * Product_{k >= n+1} (1 - x^(2*k)) = .... - Peter Bala, Jun 24 2025

Additional comments from Michael Somos, Apr 27 2000

A006950 G.f.: Product_{k>=1} (1 + x^(2k - 1)) / (1 - x^(2k)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 28, 35, 43, 55, 70, 86, 105, 130, 161, 196, 236, 287, 350, 420, 501, 602, 722, 858, 1016, 1206, 1431, 1687, 1981, 2331, 2741, 3206, 3740, 4368, 5096, 5922, 6868, 7967, 9233, 10670, 12306, 14193, 16357, 18803, 21581

Offset: 0

N. J. A. Sloane, Warren D. Smith

nonn

Also the number of partitions of n in which all odd parts are distinct. There is no restriction on the even parts. E.g., a(9)=13 because "9 = 8+1 = 7+2 = 6+3 = 6+2+1 = 5+4 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 4+2+2+1 = 3+2+2+2 = 2+2+2+2+1". - Noureddine Chair, Feb 03 2005
Number of partitions of n in which each even part occurs with even multiplicity. There is no restriction on the odd parts.
Also the number of partitions of n into parts not congruent to 2 mod 4. - James Sellers, Feb 08 2002
Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras o(n) of skew-symmetric n X n matrices, n=0,1,2,3,... (the cases n=0,1 being degenerate). This sequence, A015128 and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003
Poincaré series [or Poincare series] (or Molien series) for symmetric invariants in F_2(b_1, b_2, ... b_n) ⊗ E(e_1, e_2, ... e_n) with b_i 2-dimensional, e_i one-dimensional and the permutation action of S_n, in the case n=2.
Equals polcoeff inverse of A010054 with alternate signs. - Gary W. Adamson, Mar 15 2010
It appears that this sequence is related to the generalized hexagonal numbers (A000217) in the same way as the partition numbers A000041 are related to the generalized pentagonal numbers A001318. (See the table in comments section of A195825.) Conjecture: this is 1 together with the row sums of triangle A195836, also column 1 of A195836, also column 2 of the square array A195825. - Omar E. Pol, Oct 09 2011
Since this is also column 2 of A195825 so the sequence contains only one plateau [1, 1, 1] of level 1 and length 3. For more information see A210843. - Omar E. Pol, Jun 27 2012
Convolution of A035363 and A000700. - Vaclav Kotesovec, Aug 17 2015
Also the number of ways to stack n triangles in a valley (pointing upwards or downwards depending on row parity). - Seiichi Manyama, Jul 07 2018

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 13*x^9 + ...
G.f. = q^-1 + q^7 + q^15 + 2*q^23 + 3*q^31 + 4*q^39 + 5*q^47 + 7*q^55 + 10*q^63 + ...
From _Seiichi Manyama_, Jul 07 2018: (Start)
n | the ways to stack n triangles in a valley
--+------------------------------------------------------
1 | *---*
  |  \ /
  |   *
  |
2 |   *
  |  / \
  | *---*
  |  \ /
  |   *
  |
3 |   *---*     *---*
  |  / \ /       \ / \
  | *---*         *---*
  |  \ /           \ /
  |   *             *
  |
4 |     *                       *
  |    / \                     / \
  |   *---*     *---*---*     *---*
  |  / \ /       \ / \ /       \ / \
  | *---*         *---*         *---*
  |  \ /           \ /           \ /
  |   *             *             *
  |
5 |     *---*         *         *         *---*
  |    / \ /         / \       / \         \ / \
  |   *---*     *---*---*     *---*---*     *---*
  |  / \ /       \ / \ /       \ / \ /       \ / \
  | *---*         *---*         *---*         *---*
  |  \ /           \ /           \ /           \ /
  |   *             *             *             *
  |
6 |       *
  |      / \
  |     *---*         *---*     *   *     *---*
  |    / \ /         / \ /     / \ / \     \ / \
  |   *---*     *---*---*     *---*---*     *---*---*
  |  / \ /       \ / \ /       \ / \ /       \ / \ /
  | *---*         *---*         *---*         *---*
  |  \ /           \ /           \ /           \ /
  |   *             *             *             *
  |   *
  |  / \
  | *---*
  |  \ / \
  |   *---*
  |    \ / \
  |     *---*
  |      \ /
  |       *
(End)

A. Adem and R. J. Milgram, Cohomology of Finite Groups, Springer-Verlag, 2nd. ed., 2004; p. 108.
M. D. Hirschhorn, The Power of q, Springer, 2017. See pod, page 297.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Luca Ferrari, Schröder partitions, Schröder tableaux and weak poset patterns, arXiv:1606.06624 [math.CO], 2016. Mentions this sequence.
M. S. Mahadeva Naika and D. S. Gireesh, Arithmetic Properties of Partition k-tuples with Odd Parts Distinct, JIS, Vol. 19 (2016), Article 16.5.7
Mircea Merca, New relations for the number of partitions with distinct even parts, Journal of Number Theory 176 (July 2017), 1-12.
Victor S. Miller, Counting Matrices that are Squares, arXiv:1606.09299 [math.GR], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017. See Example 4.2 p. 13.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
L. Wang, New Congruences for Partitions where the Odd Parts are Distinct, J. Int. Seq. 18 (2015) # 15.4.2.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, arXiv preprint arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012

See also Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A015128, A046682, A106459.
Cf. A163203.
Cf. A010054, A085642, A316384.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i>n, 0, b(n-i, i-irem(i, 2)))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 06 2013

CoefficientList[ Series[ Product[(1 + x^(2k - 1))/(1 - x^(2k)), {k, 25}], {x, 0, 50}], x] (* Robert G. Wilson v, Jun 28 2012 *)
CoefficientList[Series[x*QPochhammer[-1/x, x^2] / ((1+x)*QPochhammer[x^2, x^2]), {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
CoefficientList[Series[2*(-x)^(1/8) / EllipticTheta[2, 0, Sqrt[-x]], {x, 0, 50}], x] (* Vaclav Kotesovec, Aug 17 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-Mod[i, 2]]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 11 2018, after Alois P. Heinz *)

{a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d)*x^m/m)+x*O(x^n)), n)} \\ Paul D. Hanna, Jul 22 2009
(GW-BASIC)
' A program with two A-numbers (Note that here A000217 are the generalized hexagonal numbers):
10 Dim A000217(100), A057077(100), a(100): a(0)=1
20 For n = 1 to 51: For j = 1 to n
30 If A000217(j) <= n then a(n) = a(n) + A057077(j-1)*a(n - A000217(j))
40 Next j: Print a(n-1);: Next n ' Omar E. Pol, Jun 10 2012

a(n) = (1/n)*Sum_{k=1..n} (-1)^(k+1)*A002129(k)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Feb 05 2002
G.f.: 1/Sum_{k>=0} (-x)^(k*(k+1)/2). - Vladeta Jovovic, Sep 22 2002 [corrected by Vaclav Kotesovec, Aug 17 2015]
a(n) = A059777(n-1)+A059777(n), n > 0. - Vladeta Jovovic, Sep 22 2002
G.f.: Product_{m>=1} (1+x^m)^(if A001511(m) > 1, A001511(m)-1 else A001511(m)). - Jon Perry, Apr 15 2005
Expansion of 1 / psi(-x) in powers of x where psi() is a Ramanujan theta function.
Expansion of q^(1/8) * eta(q^2) / (eta(q) * eta(q^4)) in powers of q.
Convolution inverse of A106459. - Michael Somos, Nov 02 2005
G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ). - Paul D. Hanna, Jul 22 2009
a(n) ~ (8*n+1) * cosh(sqrt(8*n-1)*Pi/4) / (16*sqrt(2)*n^2) - sinh(sqrt(8*n-1)*Pi/4) / (2*Pi*n^(3/2)) ~ exp(Pi*sqrt(n/2))/(4*sqrt(2)*n) * (1 - (2/Pi + Pi/16)/sqrt(2*n) + (3/16 + Pi^2/1024)/n). - Vaclav Kotesovec, Aug 17 2015, extended Jan 09 2017
Can be computed recursively by Sum_{j>=0} (-1)^(ceiling(j/2)) a(n - j(j+1)/2) = 0, for n > 0. [Merca, Theorem 4.3] - Eric M. Schmidt, Sep 21 2017
a(n) = A000041(n) - A085642(n), for n >= 1. - Wouter Meeussen, Dec 20 2017

G.f. and more terms from Vladeta Jovovic, Feb 05 2002

A089807 Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1

Offset: 0

Eric W. Weisstein, Nov 12 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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^5, b = x. - Michael Somos, Jul 12 2012
Number 11 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 - q - q^4 + 2*q^9 - q^16 - q^25 + 2*q^36 - q^49 - q^64 + 2*q^81 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Eric Weisstein's World of Mathematics, Quintuple Product Identity.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
I. J. Zucker, Further Relations Amongst Infinite Series and Products. II. The Evaluation of Three-Dimensional Lattice Sums, J. Phys. A: Math. Gen. 23, 117-132, 1990.

Related to the 14 primitive eta-products which are holomorphic modular forms of weight 1/2: A000122, A002448, A010054, A010815, A080995, A089801, A089802, this sequence, A089810, A089812, A106459, A121373, A133985, A133988. - Seiichi Manyama, May 15 2017

Cf. A000700, A000122, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q], {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 3, 0, q^9] - EllipticTheta[ 3, 0, q])/2, {q, 0, n}]; (* Michael Somos, Jul 12 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3, q^6] EllipticTheta[ 2, 0, Sqrt[ -q]] / (2 (-q)^(1/8)), {q, 0, n}] (* Michael Somos, Jul 12 2012 *);

{a(n) = if( n<1, n==0, issquare(n) * (3*(n%3==0) - 1))}; /* Michael Somos, Nov 05 2005 */

a(n) = -b(n) where b() is multiplicative with b(3^e) = -2(1 + (-1)^e) / 2 if e>0, b(p^e) = (1 + (-1)^e) / 2 otherwise.
From Michael Somos, Nov 05 2005: (Start)
Expansion of eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 0, 0, -1, -1, ...].
G.f.: (Sum_{k in Z} 3 * x^((3*k)^2) - x^(k^2)) / 2 = Product_{k>0} (1 - x^k) / ((1 - x^(12*k - 2)) * (1 - x^(12*k - 3)) * (1 - x^(12*k - 9)) * (1 - x^(12*k - 10))). (End)
Expansion of Jacobi theta function theta_3(Pi/3, q) in powers of q. - Michael Somos, Jan 26 2006
Expansion of chi(q^3) * psi(-q) in powers of q where chi(), psi() are Ramanujan theta functions. - Michael Somos, May 19 2007
Expansion of f(x*w, x/w) in powers of x where w is a primitive cube root of unity and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A089801.
a(n) = (-1)^n * A089810(n). - Michael Somos, Jan 20 2012
For n > 0, a(n) = (floor(sqrt(n))-floor(sqrt(n-1)))*(2-4*sin(floor(sqrt(n))*Pi/3)^2). - Mikael Aaltonen, Jan 17 2015
Sum_{k=1..n} abs(a(k)) ~ (4/3)*sqrt(n). - Amiram Eldar, Jan 27 2024

A192540 G.f.: A(x) = Series_Reversion(xG(x)) where G(x) = Sum_{n>=0} (-x)^(n(n+1)/2).

Original entry on oeis.org

1, 1, 2, 6, 20, 70, 255, 960, 3707, 14597, 58382, 236522, 968597, 4003061, 16674858, 69936760, 295092057, 1251747436, 5334958079, 22834290248, 98108081192, 422986894605, 1829443421394, 7935301625600, 34510975557383, 150456011512671, 657415433062780

Offset: 1

Paul D. Hanna, Jul 03 2011

nonn

Related q-series: Sum_{n>=0} (-q)^(n*(n+1)/2) = q^(-1/8)*eta(q)*eta(q^4)/eta(q^2) is a g.f. of A106459.

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 70*x^6 + 255*x^7 + ...
The g.f. A = A(x) satisfies the following relations:
(1) A = x/(1 - A - A^3 + A^6 + A^10 - A^15 - A^21 + A^28 + A^36 + ...).
(2) A = x/((1-A)*(1+A^2)* (1-A^2)*(1+A^4)* (1-A^3)*(1+A^6)* (1-A^4)*(1+A^8)*...).
(3) A = x/((1-A)*(1-A^4)* (1-A^3)*(1-A^8)* (1-A^5)*(1-A^12)* (1-A^7)*(1-A^16)*...).
(4) A = x*(1+A)/(1-A^2)* (1+A^3)/(1-A^4)* (1+A^5)/(1-A^6) * (1+A^7)/(1-A^8)*...
(5) A = x*(1-A^2)/(1-A)* (1-A^6)/(1-A^2)* (1-A^10)/(1-A^3)* (1-A^14)/(1-A^4)*...
(6) A = x*exp(A/(1-A) - A^2/(2*(1+A^2)) + A^3/(3*(1-A^3)) - A^4/(4*(1+A^4)) + ...).
(7) A = x*exp(A + A^2/2 + 4*A^3/3 + 5*A^4/4 + 6*A^5/5 +...+ A113184(n)*A^n/n + ...).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Cf. A106459, A006950, A296045.

nmax:=27: with(gfun): f := proc(x): x*add((-x)^(n*(n+1)/2),n=0..nmax) end: S:=series(f(x),x,nmax): g:= seriestoseries(S,'revogf'): seq(coeftayl (g,x=0,n),n=1..nmax); # Johannes W. Meijer, Jul 04 2011

Rest[CoefficientList[InverseSeries[Series[x*EllipticTheta[2, 0, Sqrt[-x]] / (2*(-x)^(1/8)), {x, 0, 30}], x], x]] (* Vaclav Kotesovec, Aug 17 2015 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, 8*(s/Sqrt[2*Pi*(77 - 8*(-s)^(7/8) *s*(Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-s]] / r))])} /. FindRoot[{2*r == -(-s)^(7/8)*EllipticTheta[2, 0, Sqrt[-s]], 2*(-s)^(11/8)*Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-s]] == 7*r}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

{a(n)=polcoeff(serreverse(x*sum(m=0,sqrtint(2*n)+1,(-x)^(m*(m+1)/2)+x*O(x^n))),n)}

{a(n)=local(A=x+x^2);for(i=1,n,A=x/prod(m=1,n,(1 - A^m)*(1 + A^(2*m))+x*O(x^n)));polcoeff(A,n)}

{a(n)=local(A=x+x^2);for(i=1,n,A=x/prod(m=1,n\2,(1 - A^(2*m-1))*(1 - A^(4*m))+x*O(x^n)));polcoeff(A,n)}

{a(n)=local(A=x+x^2);for(i=1,n,A=x*prod(m=1,n\2,(1 + A^(2*m-1))/(1 - A^(2*m)+x*O(x^n))));polcoeff(A,n)}

{a(n)=local(A=x+x^2);for(i=1,n,A=x*prod(m=1,n,(1 - A^(4*m-2))/(1 - A^m+x*O(x^n))));polcoeff(A,n)}

{a(n)=local(A=x+x^2); for(i=1, n, A=x*exp(sum(m=1, n, -(-A+x*O(x^n))^m/(1+(-A)^m)/m))); polcoeff(A, n)}

{a(n)=if(n<1,0,(1/n)*polcoeff(x/prod(k=1,n,(1-x^k)*(1+x^(2*k)+x*O(x^n)))^n,n))}

{a(n)=local(A=x+x^2);for(i=1,n,A=x*exp(sum(m=1,n, A^m*sumdiv(m,d,(-1)^(m-d)*d)/m)+x*O(x^n)));polcoeff(A,n)}

G.f. satisfies:
(1) A(x) = x/[Sum_{n>=0} (-A(x))^(n*(n+1)/2)].
(2) A(x) = x/[Product_{n>=1} (1 - A(x)^n)*(1 + A(x)^(2*n))].
(3) A(x) = x/[Product_{n>=1} (1 - A(x)^(2*n-1))*(1 - A(x)^(4*n))].
(4) A(x) = x* Product_{n>=1} (1 + A(x)^(2*n-1))/(1 - A(x)^(2*n)).
(5) A(x) = x* Product_{n>=1} (1 - A(x)^(4*n-2))/(1 - A(x)^n).
(6) A(x) = x* exp( Sum_{n>=1} -(-A(x))^n/(n*(1 + (-A(x))^n)) ).
(7) A(x) = x* exp( Sum_{n>=1} A(x)^n*Sum_{d|n} (-1)^(n-d)*d/n ).
a(n) = [x^n] (1/n)*x/[Product_{k>=1} (1 - x^k)*(1 + x^(2*k))]^n for n >= 1.
a(n) ~ c * d^n / n^(3/2), where d = 4.6257905683677649210878404538251898489748116820946869227688637924996..., c = 0.1001072494040204029591345793571534412084516176488795... . - Vaclav Kotesovec, Aug 17 2015

A215597 Expansion of psi(-x) * f(-x)^3 in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -4, 3, 4, -2, 0, -11, 4, 0, 12, 10, -12, -7, -4, 0, -12, 16, 0, 6, 0, 9, 8, -10, 0, -18, -20, 0, 20, -14, 12, 11, 24, 0, 0, -22, 0, 16, -20, -6, -12, 0, 0, -3, 4, 0, -20, 48, 0, 14, 28, 0, -40, 0, 0, 0, -8, -33, -4, -26, 0, 30, 28, 0, 0, 2, 12, -16, 20, 0

Offset: 0

Michael Somos, Aug 16 2012

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			1 - 4*x + 3*x^2 + 4*x^3 - 2*x^4 - 11*x^6 + 4*x^7 + 12*x^9 + 10*x^10 + ...
q - 4*q^5 + 3*q^9 + 4*q^13 - 2*q^17 - 11*q^25 + 4*q^29 + 12*q^37 + 10*q^41 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A010816, A106459, A215596.

A215597[n_] := SeriesCoefficient[(QPochhammer[x]^4 * QPochhammer[x^4])/ QPochhammer[x^2], {x, 0, n}]; Table[A215597[n], {n, 0, 50}] (* G. C. Greubel, Oct 01 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A) / eta(x^2 + A), n))}

Expansion of q^(-1/4) * eta(q)^4 * eta(q^4) / eta(q^2) in powers of q.
Euler transform of period 4 sequence [ -4, -3, -4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (128 t)) = 2^(19/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215596.
a(n) = (-1)^floor( n/2 ) * b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-p)^(e/2) if p == 3 (mod 4),
Convolution of A106459 and A010816.

A374080 Expansion of Product_{k>=1} (1 - x^(4k-1)) (1 - x^(4*k)).

Original entry on oeis.org

1, 0, 0, -1, -1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, -1, -1, 0, 0, 0, -1, 0, 1, 0, -1, -1, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 1, -1, -1, 1, 1, 0, 0, 1, 2, -1, -1, 0, 1, 0, -2, 0, 2, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -2, 0, 1, -1, -2, 1, 2, 0, -2, -1, 2, 0, -2

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A014601, A035364, A081362, A082303, A106459, A137569, A274719, A284316, A301505, A374058, A374060, A374081.

nmax = 85; CoefficientList[Series[Product[(1 - x^(4 k - 1)) (1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[DivisorSum[k, # &, Or[Mod[#, 4] == 0, Mod[#, 4] == 3] &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

A374081 Expansion of Product_{k>=1} (1 - x^(4k-3)) (1 - x^(4*k)).

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, 0, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 2, -1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 1, 1, 0, -2, 1, 1, -2, 0, 2, 0, -1, 0, 1, 0, -1, 0, 2, -1, -2, 1, 1, -1, -1, 1, 2, -2, -1, 2, 0, -2, 0, 2, 0, -2, 0, 2, -1, -2, 1, 2, -1, -2, 1, 2

Offset: 0

Ilya Gutkovskiy, Jun 27 2024

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000

Cf. A010815, A035362, A042948, A081362, A082303, A106459, A137569, A274719, A284313, A301504, A374058, A374060, A374080.

nmax = 85; CoefficientList[Series[Product[(1 - x^(4 k - 3)) (1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -(1/n) Sum[DivisorSum[k, # &, Or[Mod[#, 4] == 0, Mod[#, 4] == 1] &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 85}]

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^jx^(kj*(j-1)/2 + j^2).

Original entry on oeis.org

1, 1, -2, 1, -1, 0, 1, -1, -1, 0, 1, -1, 0, 0, 2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 1, 1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, -2, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0

Offset: 0

Ilya Gutkovskiy, Aug 13 2017

			Square array begins:
   1,   1,   1,   1,   1,   1, ...
  -2,  -1,  -1,  -1,  -1,  -1, ...
   0,  -1,   0,   0,   0,   0, ...
   0,   0,  -1,   0,   0,   0, ...
   2,   0,   0,  -1,   0,   0, ...
   0,   1,   0,   0,  -1,   0, ...

G. C. Greubel, Table of n, a(n) for the first 100 antidiagonals, flattened
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Columns k=0..10 give A002448, A010815, A106459, A113429, A089802, A232714, A244525, A281814 (absolute values), A205988 (absolute values), A281815 (absolute values), A247133.

Cf. A000041, A000290, A000326, A000384, A000566, A000567, A001106, A001107, A006950, A015128, A036820, A051682, A051624, A051865, A195825, A195848, A195849, A195850, A195851, A195852, A196933, A211970.

Table[Function[k, SeriesCoefficient[Sum[(-1)^i x^(k i (i - 1)/2 + i^2), {i, -n, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Product[(1 - x^((k + 2) i)) (1 - x^((k + 2) i - 1)) (1 - x^((k + 2) i - k - 1)), {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(2 + k) QPochhammer[1/x, x^(2 + k)] QPochhammer[x^(-1 - k), x^(2 + k)] QPochhammer[x^(2 + k), x^(2 + k)])/((-1 + x) (-1 + x^(1 + k))), {x, 0, n}]][j - n], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column 0: Sum_{j=-inf..inf} (-1)^j*x^A000290(j) = Product_{i>=1} (1 + x^i)/(1 - x^i) (convolution inverse of A015128).
G.f. of column 1: Sum_{j=-inf..inf} (-1)^j*x^A000326(j) = Product_{i>=1} (1 - x^i) (convolution inverse of A000041).
G.f. of column 2: Sum_{j=-inf..inf} (-1)^j*x^A000384(j) = Product_{i>=1} (1 - x^(2*i))/(1 + x^(2*i-1)) (convolution inverse of A006950).
G.f. of column 3: Sum_{j=-inf..inf} (-1)^j*x^A000566(j) = Product_{i>=1} (1 - x^(5*i))*(1 - x^(5*i-1))*(1 - x^(5*i-4)) (convolution inverse of A036820).
G.f. of column 4: Sum_{j=-inf..inf} (-1)^j*x^A000567(j) = Product_{i>=1} (1 - x^(6*i))*(1 - x^(6*i-1))*(1 - x^(6*i-5)) (convolution inverse of A195848).
G.f. of column 5: Sum_{j=-inf..inf} (-1)^j*x^A001106(j) = Product_{i>=1} (1 - x^(7*i))*(1 - x^(7*i-1))*(1 - x^(7*i-6)) (convolution inverse of A195849).
G.f. of column 6: Sum_{j=-inf..inf} (-1)^j*x^A001107(j) = Product_{i>=1} (1 - x^(8*i))*(1 - x^(8*i-1))*(1 - x^(8*i-7)) (convolution inverse of A195850).
G.f. of column 7: Sum_{j=-inf..inf} (-1)^j*x^A051682(j) = Product_{i>=1} (1 - x^(9*i))*(1 - x^(9*i-1))*(1 - x^(9*i-8)) (convolution inverse of A195851).
G.f. of column 8: Sum_{j=-inf..inf} (-1)^j*x^A051624(j) = Product_{i>=1} (1 - x^(10*i))*(1 - x^(10*i-1))*(1 - x^(10*i-9)) (convolution inverse of A195852).
G.f. of column 9: Sum_{j=-inf..inf} (-1)^j*x^A051865(j) = Product_{i>=1} (1 - x^(11*i))*(1 - x^(11*i-1))*(1 - x^(11*i-10)) (convolution inverse of A196933).
G.f. of column k: Sum_{j=-inf..inf} (-1)^j*x^(k*j*(j-1)/2+j^2) = Product_{i>=1} (1 - x^((k+2)*i))*(1 - x^((k+2)*i-1))*(1 - x^((k+2)*i-k-1)).

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A010054 a(n) = 1 if n is a triangular number, otherwise 0.

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A006950 G.f.: Product_{k>=1} (1 + x^(2*k - 1)) / (1 - x^(2*k)).

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A089807 Expansion of Jacobi theta function (3theta_3(q^9)-theta_3(q))/2.

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A192540 G.f.: A(x) = Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (-x)^(n*(n+1)/2).

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A215597 Expansion of psi(-x) * f(-x)^3 in powers of x where psi(), f() are Ramanujan theta functions.

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A006950 G.f.: Product_{k>=1} (1 + x^(2k - 1)) / (1 - x^(2k)).

A192540 G.f.: A(x) = Series_Reversion(xG(x)) where G(x) = Sum_{n>=0} (-x)^(n(n+1)/2).

A374080 Expansion of Product_{k>=1} (1 - x^(4k-1)) (1 - x^(4*k)).

A374081 Expansion of Product_{k>=1} (1 - x^(4k-3)) (1 - x^(4*k)).

A287325 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Sum_{j=-inf..inf} (-1)^jx^(kj*(j-1)/2 + j^2).