A008441 -id:A008441 - cp's OEIS frontend

A014809 Expansion of Jacobi theta constant (theta_2/2)^24.

1, 24, 276, 2048, 11178, 48576, 177400, 565248, 1612875, 4200352, 10131156, 22892544, 48897678, 99448320, 193740408, 363315200, 658523925, 1157743824, 1980143600, 3303168000, 5386270686, 8602175744, 13477895856, 20748607488, 31425764410, 46883528256, 68969957700

Offset: 0

N. J. A. Sloane

nonn

Number of ways of writing n as the sum of 24 triangular numbers from A000217.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
James G. Huard and Kenneth S. Williams, Sums of sixteen and twenty-four triangular numbers, Rocky Mountain J. Math. Volume 35, Number 3 (2005), 857-868.
Ken Ono, Sinai Robins and Patrick T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=24, Theorem 8; alternative link.

Column k=24 of A286180.
Cf. A000217, A000594, A096963.
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.

a[n_] := Module[{e = IntegerExponent[n+3, 2]}, (2^(11*e) * DivisorSigma[11, (n+3)/2^e] - RamanujanTau[n+3] - 2072 * If[OddQ[n], RamanujanTau[(n+3)/2], 0]) / 176896]; Array[a, 27, 0] (* Amiram Eldar, Jan 11 2025 *)

From Wolfdieter Lang, Jan 13 2017: (Start)
G.f.: 24th power of the g.f. for A010054.
a(n) = (A096963(n+3) - tau(n+3) - 2072*tau((n+3)/2))/176896, with Ramanujan's tau function given in A000594, and tau(n) is put to 0 if n is not integer. See the Ono et al. link, case k=24, Theorem 8. (End)
a(n) = 1/72 * Sum_{a, b, x, y > 0, a*x + b*y = n + 3, x == y == 1 mod 2 and a > b} (a*b)^3*(a^2 - b^2)^2. - Seiichi Manyama, May 05 2017
a(0) = 1, a(n) = (24/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 24*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

More terms from Seiichi Manyama, May 05 2017

A113407 Expansion of psi(x) * phi(x^2) in powers of x where psi(), phi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 3, 0, 2, 1, 0, 4, 2, 1, 2, 2, 0, 2, 1, 0, 2, 4, 2, 0, 3, 0, 4, 2, 0, 0, 0, 3, 2, 2, 0, 2, 4, 0, 2, 3, 0, 4, 2, 0, 0, 2, 0, 2, 1, 2, 4, 0, 0, 2, 2, 0, 6, 2, 1, 2, 2, 0, 0, 4, 0, 0, 4, 0, 2, 1, 0, 4, 0, 0, 2, 2, 4, 2, 2, 0, 2, 5, 0, 2, 0, 2, 0, 2, 0, 4, 4, 0, 0, 0, 1, 0, 4, 0, 2, 2, 0, 4, 4, 2, 2, 0, 0, 2

Offset: 0

Michael Somos, Oct 28 2005

nonn

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

Bisection of A008441. Number of ways to write n as two times a square plus a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 2*x^5 + x^6 + 4*x^8 + 2*x^9 + x^10 + 2*x^11 + ...
G.f. = q + q^9 + 2*q^17 + 3*q^25 + 2*q^41 + q^49 + 4*q^65 + 2*q^73 + q^81 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(vi).

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441, A053692.

phi[x_] := EllipticTheta[3, 0, x]; psi[x_] := (1/2)*x^(-1/8)*EllipticTheta[2, 0, x^(1/2)]; s = Series[psi[x]*phi[x^2], {x, 0, 104}]; a[n_] := Coefficient[s, x, n] ; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 17 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^5 / (eta(x + A) * eta(x^8 + A)^2), n))};

Expansion of chi(x) * f(x)^2 in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Jul 24 2012
Expansion of q^(-1/8) * eta(q^4)^5 / (eta(q) * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ 1, 1, 1, -4, 1, 1, 1, -2, ...].
a(n) = b(8*n + 1), where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 +(-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jul 24 2012
G.f.: (Sum_{k in Z} x^(2*k^2)) * (Sum_{k>=0} x^((k^2 + k)/2)) = Sum_{k>=0} (-1)^k * (x^(2*k + 1) + 1) / (x^(2*k + 1) - 1) * x^((k^2 + k)/2).
a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = a(n). a(n) = A008441(2*n). - Michael Somos, Jul 24 2012

A035154 a(n) = Sum_{d|n} Kronecker(-36, d).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 0, 1, 1, 2, 0, 1, 2, 0, 2, 1, 2, 1, 0, 2, 0, 0, 0, 1, 3, 2, 1, 0, 2, 2, 0, 1, 0, 2, 0, 1, 2, 0, 2, 2, 2, 0, 0, 0, 2, 0, 0, 1, 1, 3, 2, 2, 2, 1, 0, 0, 0, 2, 0, 2, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 3, 0, 0, 2, 0, 2, 1, 2, 0, 0, 4, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 1, 2, 1, 0, 3, 2, 2, 0, 2, 0

Offset: 1

N. J. A. Sloane

			G.f. = x + x^2 + x^3 + x^4 + 2*x^5 + x^6 + x^8 + x^9 + 2*x^10 + x^12 + 2*x^13 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, 1994, see p. 197, Entry 44.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002654, A008441, A019670, A122856, A122857, A122865, A125079.

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 24 2011 *)
a[ n_] := SeriesCoefficient[ (-2 + EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 4, {q, 0, n}]; (* Michael Somos, Jul 09 2013 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -36, d)))}; /* Michael Somos, Jul 30 2006 */

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -36, p) * X))) [n])}; /* Michael Somos, Jul 30 2006 */

{a(n)=polcoeff(sum(m=0,n\6+1,(-1)^m*(x^(6*m+1)/(1-x^(6*m+1)+x*O(x^n)) + x^(6*m+5)/(1-x^(6*m+5)+x*O(x^n)))),n)} /* Paul D. Hanna */

Expansion of -1 + (theta_3(q)^2 + theta_3(q^3)^2) / 2 in powers of q. - Michael Somos, Jul 09 2013
From Michael Somos, Jul 30 2006: (Start)
Moebius transform is period 12 sequence [1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, ...].
Multiplicative with a(2^e) = a(3^e) = 1, a(p^e) = e+1 if p == 1(mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3(mod 4). (End)
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -36, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker( -36, p) * p^-s)). - Michael Somos, Jun 24 2011
a(2*n) = a(3*n) = a(n). a(2*n + 1) = A125079(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n).
2 * a(n) = A122857(n) unless n=0. - Michael Somos, Jul 09 2013
G.f.: Sum_{n>=0} (-1)^n*( x^(6*n+1)/(1-x^(6*n+1)) + x^(6*n+5)/(1-x^(6*n+5)) ). - Paul D. Hanna, Dec 14 2011
G.f.: x/(1-x) + x^5/(1-x^5) - x^7/(1-x^7) - x^11/(1-x^11) + x^13/(1-x^13) + x^17/(1-x^17) --++ ...
a(n) = A002654(n) + A002654(3*n). - Michael Somos, Jan 25 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Nov 17 2023

A104794 Expansion of theta_4(q)^2 in powers of q.

Original entry on oeis.org

1, -4, 4, 0, 4, -8, 0, 0, 4, -4, 8, 0, 0, -8, 0, 0, 4, -8, 4, 0, 8, 0, 0, 0, 0, -12, 8, 0, 0, -8, 0, 0, 4, 0, 8, 0, 4, -8, 0, 0, 8, -8, 0, 0, 0, -8, 0, 0, 0, -4, 12, 0, 8, -8, 0, 0, 0, 0, 8, 0, 0, -8, 0, 0, 4, -16, 0, 0, 8, 0, 0, 0, 4, -8, 8, 0, 0, 0, 0, 0, 8

Offset: 0

Michael Somos, Mar 26 2005

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Quadratic AGM theta functions: a(q) (see A004018), b(q) (A104794), c(q) (A005883).
In the Arithmetic-Geometric Mean, if a = theta_3(q)^2, b = theta_4(q)^2 then a' := (a+b)/2 = theta_3(q^2)^2, b' := sqrt(a*b) = theta_4(q^2)^2.

			G.f. = 1 - 4*q + 4*q^2 + 4*q^4 - 8*q^5 + 4*q^8 - 4*q^9 + 8*q^10 - 8*q^13 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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].
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000203, A000729, A001934, A002131, A004018, A008441, A053692, A113407, A113652, A122856, A122865, A204531, A227695, A246950, A256014, A258210.

# JacobiTheta4 is defined in A002448.
A104794List(len) = JacobiTheta4(len, 2)
A104794List(102) |> println # Peter Luschny, Mar 12 2018

A := Basis( ModularForms( Gamma1(8), 1), 100); A[1] - 4*A[2] + 4*A[3]; /* Michael Somos, Jan 31 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2, {q, 0, n}];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[1 - m] EllipticK[m] / (Pi/2), {q, 0, n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(1/4) EllipticK[m] / (Pi/2), {q, 0, 2 n}]];
a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n 4 DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jun 06 2015 *)

{a(n) = if( n<1, n==0, (-1)^n * 4 * sumdiv(n, d, (d%4==1) - (d%4==3)))};

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

{a(n) = if( n<0, 0, polcoeff( 1 + 4 * sum( k=1, n, (-x)^k / (1 + x^(2*k)), x * O(x^n)), n))};

Expansion of phi(-q)^2 = 2 * phi(q^2)^2 - phi(q)^2 = (phi(q) - 2*phi(q^4))^2 = f(-q)^3 / psi(q) = phi(-q^2)^4 / phi(q)^2 = psi(-q)^4 / psi(q^2)^2 = psi(q)^2 * chi(-q)^6 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (1-k^2)^(1/2) K(k^2) / (Pi/2) in powers of q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind.
Expansion of  K(k^2) / (Pi/2) in powers of -q where q is Jacobi's nome, k is the elliptic modulus and K() is the complete elliptic integral of the first kind. - Michael Somos, Jun 08 2015
Expansion of eta(q)^4 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -4, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u^2 + v^2) - 2*u*w^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 - 2*u1*u3 + 4*u2*u6 - 3*u3^2.
Moebius transform is period 8 sequence [ -4, 8, 4, 0, -4, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 16 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008441.
G.f.: theta_4(q)^2 = (Sum_{k in Z} (-q)^(k^2))^2 = (Product_{k>0} (1 - q^(2*k)) * (1 - q^(2*k - 1))^2)^2.
G.f.: 1 + 4 * Sum_{k>0} (-x)^k / (1 + x^(2*k)). - Michael Somos, Jun 08 2015
a(4*n + 3) = 0. a(n) = (-1)^n * A004018(n) = a(2*n). a(4*n + 1) = -4 * A008441(n). a(n) = -4 * A113652(n) unless n=0. a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = 4 * A122856(n). a(8*n + 1) = -4 * A113407(n). a(8*n + 5) = -8 * A053692(n).
a(n) = a(9*n) = A204531(8*n) = A246950(8*n) = A256014(9*n) = A258210(n). - Michael Somos, Jun 08 2015
Convolution inverse of A001934. Convolution with A000729 is A227695. - Michael Somos, Jun 08 2015
G.f.: 2 * Sum_{k in Z} (-1)^k * x^(k*(k + 1)/2) / (1 + x^k). - Michael Somos, Nov 05 2015
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0. - Seiichi Manyama, May 02 2017
G.f.: exp(2*Sum_{k>=1} (sigma(k) - sigma(2*k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A226255 Number of ways of writing n as the sum of 11 triangular numbers.

Original entry on oeis.org

1, 11, 55, 176, 440, 957, 1848, 3245, 5412, 8580, 12892, 18888, 26895, 36916, 50160, 66935, 86658, 111870, 142582, 177320, 221100, 272690, 329065, 399102, 480040, 566808, 672969, 793760, 920326, 1074040, 1248412, 1425974, 1640595, 1882145, 2123385, 2418339, 2743928, 3062895, 3453978, 3880855

Offset: 0

N. J. A. Sloane, Jun 01 2013

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.

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.

G.f. is 11th power of g.f. for A010054.
a(0) = 1, a(n) = (11/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
G.f.: exp(Sum_{k>=1} 11*(x^k/k)/(1 + x^k)). - Ilya Gutkovskiy, Jul 31 2017

A307597 Number of partitions of n into 2 distinct positive triangular numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 3, 0, 1

Offset: 0

Ilya Gutkovskiy, Apr 17 2019

The greedy inverse (positions of first occurrence of n) starts 0, 4, 16, 81, 471, 2031, 1381, 11781, 6906, 17956, ... - R. J. Mathar, Apr 28 2020

			a(16) = 2 because we have [15, 1] and [10, 6].

Alois P. Heinz, Table of n, a(n) for n = 0..65536 (first 10000 terms from David A. Corneth)

Cf. A000217, A008441, A024940, A025441, A052344, A053603, A260647, A307598.

Cf. A010054.

a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k*(k+1)/2)).

a(n) = Sum_{k=1..floor((n-1)/2)} c(k) * c(n-k), where c = A010054. - Wesley Ivan Hurt, Jan 06 2024

A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x(x+1)/2 + y(y+1)/2 + z*(z+1)/2 is a triangular number.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 2, 3, 2, 2, 6, 1, 3, 5, 1, 2, 3, 5, 2, 1, 3, 3, 3, 4, 3, 8, 2, 5, 11, 2, 5, 8, 4, 6, 4, 9, 4, 6, 5, 4, 6, 3, 8, 8, 5, 8, 10, 7, 7, 11, 8, 6, 7, 8, 5, 9, 7, 6, 8, 7, 7, 8, 13, 9, 11, 10, 7, 22, 9, 10, 13, 3, 6, 10, 8, 17, 12, 7, 9, 10, 16, 6, 18, 18, 10, 15, 9, 12, 20, 5

Offset: 1

Zhi-Wei Sun, Oct 10 2013

nonn

Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 5, 7, 12. Moreover, for each n = 20, 21, ... there are three distinct positive integers x, y and z with x + y + z = n such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.
(ii) A positive integer n cannot be written as x + y + z (x, y, z > 0) with x^2 + y^2 + z^2 a square if and only if n has the form 2^r*3^s or the form 2^r*7, where r and s are nonnegative integers.
(iii) Any integer n > 14 can be written as a + b + c + d, where a, b, c, d are positive integers with a^2 + b^2 + c^2 + d^2 a square. If n > 20 is not among 22, 28, 30, 38, 44, 60, then we may require additionally that a, b, c, d are pairwise distinct.
(iv) For each integer n > 50 not equal to 71, there are positive integers a, b, c, d with a + b + c + d = n such that both a^2 + b^2 and c^2 + d^2 are squares.
Part (ii) and the first assertion in part (iii) were confirmed by Chao Huang and Zhi-Wei Sun in 2021. - Zhi-Wei Sun, May 09 2021

			a(16) = 1 since 16 = 3 + 6 + 7 and 3*4/2 + 6*7/2 + 7*8/2 = 55 = 10*11/2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..3000
Chao Huang and Zhi-Wei Sun, On partitions of integers with restrictions involving squares, arXiv:2105.03416 [math.NT], 2021.
Zhi-Wei Sun, Diophantine problems involving triangular numbers and squares, a message to Number Theory List, Oct. 11, 2013.

Cf. A000217, A000290.

Cf. A008443, A053604, A053603, A052343, A052344, A008441.

SQ[n_]:=IntegerQ[Sqrt[n]]
T[n_]:=n(n+1)/2
a[n_]:=Sum[If[SQ[8(T[i]+T[j]+T[n-i-j])+1],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

a(n)=my(t=(n+1)*n/2,s);sum(x=1,n\3,s=t-n--*x;sum(y=x,n\2,is_A000217(s-(n-y)*y))) \\ - M. F. Hasler, Oct 11 2013

A274621 Coefficients in the expansion Product_{ n>=1 } (1-q^(2n-1))^2/(1-q^(2n))^2.

Original entry on oeis.org

1, -2, 3, -6, 11, -18, 28, -44, 69, -104, 152, -222, 323, -460, 645, -902, 1254, -1722, 2343, -3174, 4278, -5722, 7601, -10056, 13250, -17358, 22623, -29382, 38021, -48984, 62857, -80404, 102528, -130282, 165002, -208398, 262495, -329666, 412878, -515840, 642941, -799362, 991478

Offset: 0

N. J. A. Sloane, Jul 03 2016

This is the reciprocal of the g.f. for A008441.
From Wolfdieter Lang, Jul 05 2016: (Start)
The g.f. is the square of the one for A106507.
Expansion of 1/(k/(4*q^(1/2)) * (2/Pi)*K(k)) in powers of q^2, where k is the modulus (k^2 is the parameter), K is the real quarter period and q is the Jacobi nome of elliptic functions. See a similar Jul 05 2016 comment on A008441. This appears as a factor in the sn and cn formulas of Abramowitz-Stegun. p. 575, 16.23.1 and 16.23.2. (End)

R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.23.1 and 16.23.2.
R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts

If the signs are deleted we get A273225.

Cf. A008441, A010054, A106507.

nmax = 40; CoefficientList[Series[Product[(1 - x^(2*k-1))^2 / (1 - x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 05 2016 *)

From Wolfdieter Lang, Jul 05 2016: (Start)
G.f.: 1/(theta_2(0, sqrt(q))/(2*q^(1/8)))^2, with the Jacobi theta_2 function.
G.f.: 1/(Sum_{n >= 0} q^(n*(n+1)/2))^2.
G.f.: 1/(Prod_{n >= 1} (1 - q^n) * (1 + q^n)^2)^2 = 1/(Prod_{n >= 1} (1 - q^(2*n)) * (1 + q^n ))^2 = Prod_{n >= 1} (1 - q^(2n-1))^2 / (1 - q^(2n))^2. For the last equality, giving the g.f. of the name, see the Euler identity, mentioned in a Jul 05 2016 comment of A010054. (End)
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(5/2)*n^(5/4)). - Vaclav Kotesovec, Jul 05 2016

A113446 Expansion of (phi(q)^2 - phi(q^3)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -1, 1, 2, -1, 0, 1, 1, 2, 0, -1, 2, 0, -2, 1, 2, 1, 0, 2, 0, 0, 0, -1, 3, 2, -1, 0, 2, -2, 0, 1, 0, 2, 0, 1, 2, 0, -2, 2, 2, 0, 0, 0, 2, 0, 0, -1, 1, 3, -2, 2, 2, -1, 0, 0, 0, 2, 0, -2, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, -3, 0, 0, -2, 0, 2, 1, 2, 0, 0, 4, 0, -2, 0, 2, 2, 0, 0, 0, 0, 0, -1, 2, 1, 0, 3, 2, -2, 0, 2, 0

Offset: 1

Michael Somos, Nov 02 2005

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

			G.f. = q + q^2 - q^3 + q^4 + 2*q^5 - q^6 + q^8 + q^9 + 2*q^10 - q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A008441, A019673, A035154, A138949.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 106); A[2] + A[3] - A[4] + A[5]; /* Michael Somos, Jan 31 2015 */

a[ n_] := If[ n < 1, 0, (-1)^IntegerExponent[ n, 3] Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^(3/2)]^3 / EllipticTheta[ 2, 0, q^(1/2)] (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3]), {q, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = if( n<1, 0, (-1)^valuation(n, 3) * sumdiv(n, d, kronecker(-36, d)))};

{a(n) = if( n<1, 0, direuler(p=2, n, if( p==3, 1 / (1 + X), 1 / (1 - X) / (1 - kronecker(-36, p) * X)))[n])};

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2), n))};

Expansion of (eta(q^2)^3 * eta(q^6) * eta(q^12)^2) / (eta(q) * eta(q^3) * eta(q^4)^2) in powers of q.
Euler transform of period 12 sequence [1, -2, 2, 0, 1, -2, 1, 0, 2, -2, 1, -2, ...].
Moebius transform is period 12 sequence [1, 0, -2, 0, 1, 0, -1, 0, 2, 0, -1, 0, ...].
a(n) is multiplicative and a(2^e) = 1, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
G.f.: ((Sum_{k} x^(k^2))^2 - (Sum_{k} x^(3*k^2))^2) / 4.
G.f.: Sum_{k>0} x^(3*k-1) / (1 + x^(6*k-2)) + x^(3*k-2)/(1 + x^(6*k-4)).
G.f.: Sum_{k>0} x^k * (1 - x^(2*k))^2 / (1 + x^(6*k)).
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^k)^3 * (1 + x^(3*k)) * (1 + x^(4*k) + x^(8*k))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138949.
a(n) = (-1)^e * A035154(n) where 3^e is the highest power of 3 dividing n.
a(4*n + 1) = A008441(n).
Expansion of q * f(-q, -q^5) * f(q, q^5)^2 / phi(-q^3) in powers of q where phi(), f(,) are Ramanujan theta functions. - Michael Somos, Jan 31 2015
Expansion of q * (psi(q^3)^3 / psi(q)) * (phi(q) / phi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/6 (A019673). - Amiram Eldar, Nov 24 2023

A125079 Excess of number of divisors of 2n+1 of form 12k+1, 12k+5 over those of form 12k+7, 12k+11.

Original entry on oeis.org

1, 1, 2, 0, 1, 0, 2, 2, 2, 0, 0, 0, 3, 1, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1, 2, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 3, 0, 0, 1, 0, 4, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 2, 2, 0, 2, 0, 1, 2, 4, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 1, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 2, 4, 0, 0, 0, 2, 4, 2, 0, 0, 0, 4, 0, 0

Offset: 0

Michael Somos, Nov 18 2006

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

			1 + x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^12 + x^13 + 2*x^14 + ...
q + q^3 + 2*q^5 + q^9 + 2*q^13 + 2*q^15 + 2*q^17 + 3*q^25 + q^27 + 2*q^29 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.56).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A008441, A019670, A035154, A121444.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ m]}]]]

{a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -36, d)))}

{a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( 6, d) * (-1)^(d\12)))}

{a(n) = if( n<0, 0, if( n%6==1, n\=3, 1); sumdiv( 2*n + 1, d, kronecker( -4, d)) )}

{a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==3, 1, if( p%4==1, e+1, !(e%2)))))))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}

a008441(n) = sumdiv(4*n+1, d, (-1)^(d\2));
a(n) = if(n%2==0, a008441(n/2), if(n%6==1, a008441((n-1)/6))); \\ Seiichi Manyama, Oct 11 2024

Expansion of q^(-1/2) * eta(q^3)^3 *eta(q^4) * eta(q^12) / (eta(q) * eta(q^6)^2) in powers of q.
Expansion of phi(-q^3) * psi(-q^3) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 1, 1, -2, 0, 1, 0, 1, 0, -2, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138745.
a(6*n + 3) = a(6*n + 5) = 0. a(6*n) = A002175(n). a(6*n + 1) = a(2*n) = A008441(n). a(6*n + 2) = 2 * A121444(n). a(n) = A035154(2*n + 1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Dec 28 2023
a(6*n + 3) = a(6*n + 5) = 0. a(6*n + 1) = A008441(n). a(6*n + k) = A008441(3*n + k/2) for k=0,2,4. - Seiichi Manyama Oct 11 2024

cp's OEIS Frontend

A014809 Expansion of Jacobi theta constant (theta_2/2)^24.

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A113407 Expansion of psi(x) * phi(x^2) in powers of x where psi(), phi() are Ramanujan theta functions.

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A035154 a(n) = Sum_{d|n} Kronecker(-36, d).

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A104794 Expansion of theta_4(q)^2 in powers of q.

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A226255 Number of ways of writing n as the sum of 11 triangular numbers.

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A307597 Number of partitions of n into 2 distinct positive triangular numbers.

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A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x*(x+1)/2 + y*(y+1)/2 + z*(z+1)/2 is a triangular number.

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A230121 Number of ways to write n = x + y + z (0 < x <= y <= z) such that x(x+1)/2 + y(y+1)/2 + z*(z+1)/2 is a triangular number.