A033761 -id:A033761 - cp's OEIS frontend

A224326 Number of partitions of n into 3 distinct triangular numbers.

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

Offset: 0

Alex Ratushnyak, Apr 03 2013

nonn

Indices of zeros: 0 followed by A002243.

T. D. Noe, Table of n, a(n) for n = 0..10000
Jon Maiga, Computer-generated formulas for A224326, Sequence Machine.

Cf. A000217, A002243, A033761.
Cf. A025436 (number of partitions of n into 3 distinct squares).
Cf. A002636 (allows nondistinct triangular numbers).

nn = 150; tri = Table[n*(n + 1)/2, {n, 0, nn}]; t = Table[0, {tri[[-1]]}]; Do[s = tri[[i]] + tri[[j]] + tri[[k]]; If[s <= tri[[-1]], t[[s]]++], {i, nn}, {j, i + 1, nn}, {k, j + 1, nn}]; t = Join[{0}, t] (* T. D. Noe, Apr 05 2013 *)

TOP = 777
for n in range(TOP):
  k = 0
  for x in range(TOP):
    s = x*(x+1)//2
    if s>n: break
    for y in range(x+1,TOP):
        sy = s + y*(y+1)//2
        if sy>n: break
        for z in range(y+1,TOP):
          sz = sy + z*(z+1)//2
          if sz>n: break
          if sz==n: k+=1
  print(str(k), end=',')

A244543 Expansion of phi(q^2) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 3, 2, 3, 0, 2, 0, 3, 3, 4, 2, 2, 0, 0, 0, 3, 2, 5, 2, 4, 0, 2, 0, 2, 1, 4, 4, 0, 0, 0, 0, 3, 4, 6, 0, 5, 0, 2, 0, 4, 2, 0, 2, 2, 0, 0, 0, 2, 1, 7, 4, 4, 0, 4, 0, 0, 4, 4, 2, 0, 0, 0, 0, 3, 0, 4, 2, 6, 0, 0, 0, 5, 2, 4, 2, 2, 0, 0, 0, 4, 5, 6, 2, 0, 0, 2

Offset: 0

Michael Somos, Jun 29 2014

nonn

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

			G.f. = 1 + q + 3*q^2 + 2*q^3 + 3*q^4 + 2*q^6 + 3*q^8 + 3*q^9 + 4*q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A010054, A033761, A112603, A113411, A121373, A244540.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + A[2] + 3*A[3];

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {1, 2, 1, 0, -1, -2, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, 1, 0, -1, -2, -1][d%8 + 1]))};

{a(n) = my(A, B); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); B = subst(A, x, x^2); polcoeff( B * (A + B) / 2, n))};

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + A[1] + 3*A[2];

Expansion of f(-q^3, -q^5)^2 * phi(q^2) / psi(-q) = f(-q^3, -q^5)^2 * chi(q^2)^2 / chi(-q) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [1, 2, -1, -2, -1, 2, 1, -2, ...].
Moebius transform is period 8 sequence [1, 2, 1, 0, -1, -2, -1, 0, ...].
a(2*n) = A244540(n). a(2*n + 1) = A113411(n). a(8*n + 1) = A112603(n). a(8*n + 3) = 2* A033761(n). a(8*n + 5) = a(8*n + 7) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*(sqrt(2)+1)/4 = 1.896118... . - Amiram Eldar, Jun 08 2025

A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -2, 1, 4, -2, 0, 1, -1, 4, -2, -2, 4, 0, 0, 1, 2, -1, -2, 4, 0, -2, 0, -2, 5, 4, -4, 0, 4, 0, 0, 1, -4, 2, 0, -1, 4, -2, 0, 4, 2, 0, -2, -2, 4, 0, 0, -2, 1, 5, -4, 4, 4, -4, 0, 0, -4, 4, -2, 0, 4, 0, 0, 1, 8, -4, -2, 2, 0, 0, 0, -1, 2, 4, -2, -2, 0, 0

Offset: 1

Michael Somos, Jun 30 2014

sign

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

			G.f. = q + q^2 - 2*q^3 + q^4 + 4*q^5 - 2*q^6 + q^8 - q^9 + 4*q^10 - 2*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A107635, A121373, A143259, A243747, A244560.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] + A[3];

a[ n_] := If[ n < 1, 0, Sum[ {1, 0, -3, 0, 3, 0, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 0, -3, 0, 3, 0, -1][d%8 + 1]))};

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

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] + A[2];

Expansion of q * f(-q, -q^7)^2 * phi(q) / psi(-q) = q * f(-q, -q^7)^2 * chi(q)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [1, -3, 3, 0, 3, -3, 1, -2, ...].
Moebius transform is period 8 sequence [1, 0, -3, 0, 3, 0, -1, 0, ...].
Convolution product of A244560 and A107635. Convolution product of A000122 and A143259.
a(n) = (A004018(n) - A033715(n)) / 2 = A243747(2*n).
a(2*n) = a(n). a(8*n + 3) = -2 * A033761(n). a(8*n + 5) = 4 * A053692(n). a(8*n + 7) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 - 1/sqrt(2))/2 = 0.460075... . - Amiram Eldar, Jun 08 2025

A245572 Expansion of phi(q) * phi(q^2) + 2 * phi(-q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

3, 2, -2, 4, 6, 0, -4, 0, 6, 6, 0, 4, 12, 0, 0, 0, 6, 4, -6, 4, 0, 0, -4, 0, 12, 2, 0, 8, 0, 0, 0, 0, 6, 8, -4, 0, 18, 0, -4, 0, 0, 4, 0, 4, 12, 0, 0, 0, 12, 2, -2, 8, 0, 0, -8, 0, 0, 8, 0, 4, 0, 0, 0, 0, 6, 0, -8, 4, 12, 0, 0, 0, 18, 4, 0, 4, 12, 0, 0, 0, 0

Offset: 0

Michael Somos, Jul 25 2014

sign

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

			G.f. = 3 + 2*q - 2*q^2 + 4*q^3 + 6*q^4 - 4*q^6 + 6*q^8 + 6*q^9 + 4*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A033715, A033761, A112603, A113411, A226240.

A := Basis( ModularForms( Gamma1(32), 1), 81);  3*A[1] + 2*A[2] - 2*A[3] + 4*A[4] + 6*A[5] - 4*A[7] + 6*A[9] + 6*A[10] + 4*A[12] + 12*A[13] + 4*A[16];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] + 2 EllipticTheta[ 3, 0, -q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}];

{a(n) = my(A, p, e); if( n<1, 3*(n==0), A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, if( e>1, 3, -1), p%8>3, (1 + (-1)^e) / 2, e+1)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^3 / (eta(x + A)^2 * eta(x^8 + A)^2) + 2 * eta(x^2 + A)^2 * eta(x^8 + A)^5 / (eta(x^4 + A)^3 * eta(x^16 + A)^2), n))};

a(n) = 2 * b(n) where b(n) is multiplicative with b(2) = -1, b(2^e) = 3 if e>1, b(p^e) = e+1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8).
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226240.
a(2*n + 1) = 2 * A113411(n). a(4*n) = 3 * A033715(n). a(8*n + 1) = 2 * A112603(n). a(8*n = 3) = 4 * A033761(n). a(8*n + 5) = a(8*n = 7) = 0.

A255258 Expansion of q^2 * phi(q) * psi(q^16) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 2

Michael Somos, Feb 19 2015

nonn

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

			G.f. = q^2 + 2*q^3 + 2*q^6 + 2*q^11 + 3*q^18 + 2*q^19 + 2*q^22 + 4*q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 2..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033761, A113411, A224609, A227395.

A := Basis( ModularForms( Gamma1(32), 1), 89); A[3] + 2*A[4] + 2*A[7] + 2*A[12];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^8] / 2, {q, 0, n}];

{a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^32 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^16 + A)), n))};

Expansion of eta(q^2)^5 * eta(q^32)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^16)) in powers of q.
Euler transform of period 32 sequence [ 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A224609.
(-1)^n * a(n) = A227395(n).
a(4*n) = a(4*n + 1) = a(8*n + 7) = 0. a(4*n + 2) = A113411(n). a(8*n + 3) = 2 * A033761(n).

A027414 G.f. for Moebius transform is x * (1 + x) / (1 + x^4).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

			x + 2*x^2 + x^3 + 2*x^4 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + x^11 + x^12 + ...

N. J. A. Sloane, Transforms

a[ n_] := If[ n < 1, 0, Sum[ {1, 1, 0, 0, -1, -1, 0, 0} [[ Mod[d, 8, 1]]], {d, Divisors @ n}]] (* Michael Somos, Nov 16 2011 *)

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -4, (d-1)%8\2 + 1)))} /* Michael Somos, Sep 20 2005 */

Moebius transform is period 8 sequence [1, 1, 0, 0, -1, -1, 0, 0, ...]. - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^k * (1 + x^k) / (1 + x^(4*k)). - Michael Somos, Sep 20 2005
a(8*n + 5) = 0. a(8*n + 3) = A033761(n). - Michael Somos, Nov 16 2011

A028572 Expansion of theta_3(z)theta_3(2z) + theta_2(z)theta_2(2z) in powers of q^(1/4).

Original entry on oeis.org

1, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 4, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 6, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 12, 2, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

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

			1 + 4*x^3 + 2*x^4 + 2*x^8 + 4*x^11 + 4*x^12 + 2*x^16 + 4*x^19 + 4*x^24 + ...
1 + 4*q^(3/4) +2*q +2*q^2 +4*q^(11/4) +4*q^3 +2*q^4 + 4*q^(19/4) +4*q^6 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033715, A033761.

terms = 105; max = Sqrt[terms] // Ceiling; s = Sum[x^(3*(n^2 + m^2) + 2*n*m), {n, -max, max}, {m, -max, max}]; CoefficientList[s, x][[1 ;; terms]] (* Jean-François Alcover, Dec 03 2015, using 2nd g.f. *)

{a(n) = if( n<1, n==0, qfrep( [3, 1; 1, 3], n)[n] * 2)} /* Michael Somos, Nov 20 2006 */

{a(n) = if( n<1, n==0, if( n%4==1 || n%4==2, 0, 2 * sumdiv( n, d, kronecker( -2, d))))} /* Michael Somos, Mar 23 2012 */

Expansion of phi(x^4) * phi(x^8) + 4 * x^3 * psi(x^8) * psi(x^16) in powers of x where phi(), psi() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) (t/i) f(t) where q = exp(2 pi i t). - Michael Somos, Mar 23 2012
G.f.: Sum_{n,m} x^(3*(n^2 + m^2) + 2*n*m). - Michael Somos, Nov 20 2006
a(4*n + 1) = a(4*n + 2) = a(8*n + 7) = 0. a(4*n) = A033715(n). a(8*n + 3) = 4 * A033761(n). - Michael Somos, Mar 23 2012

A129438 Expansion of (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 14 2007

nonn

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

			G.f. = 1 + q + 2*q^3 + 2*q^4 + 2*q^8 + 3*q^9 + 2*q^11 + 4*q^12 + 2*q^16 + ...

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. A033715, A033761, A112603, A113411, A125096.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] + EllipticTheta[ 4, 0, q^2] EllipticTheta[ 3, 0, q^4]) / 2, {q, 0, n}]; (* Michael Somos, Nov 11 2015 *)

{a(n) = if( n<1, n==0, qfrep([1, 0; 0, 8], n)[n] + qfrep([3, 1; 1, 3], n)[n])};

Moebius transform is period 32 sequence [1, -1, 1, 2, -1, -1, -1, 0, 1, 1, 1, 2, -1, 1, -1, 0, 1, -1, 1, -2, -1, -1, -1, 0, 1, 1, 1, -2, -1, 1, -1, 0, ...].
a(4*n + 2) = a(8*n + 5) = a(8*n + 7) = 0.
a(n) = A125096(n) unless n=0. a(8*n + 1) = A112603(n). a(8*n + 3) = 2 * A033761(n).
a(2*n + 1) = A113411(n). a(4*n) = A033715(n). - Michael Somos, Nov 11 2015

A244544 Expansion of (phi(q) + phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 2, 0, 3, 4, 4, 2, 2, 2, 0, 0, 3, 4, 5, 2, 4, 0, 2, 0, 2, 4, 4, 4, 0, 2, 0, 0, 3, 4, 6, 0, 5, 2, 2, 0, 4, 4, 0, 2, 2, 2, 0, 0, 2, 2, 7, 4, 4, 2, 4, 0, 0, 4, 4, 2, 0, 2, 0, 0, 3, 4, 4, 2, 6, 0, 0, 0, 5, 4, 4, 2, 2, 0, 0, 0, 4, 6, 6, 2, 0, 4, 2

Offset: 0

Michael Somos, Jun 29 2014

nonn

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

			G.f. = 1 + 2*q + 3*q^2 + 2*q^3 + 3*q^4 + 2*q^5 + 2*q^6 + 3*q^8 + 4*q^9 + ...

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

Cf. A033761, A053692, A093709, A244540.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 2*A[2] + 3*A[3];

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {2, 1, 0, 0, 0, -1, -2, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 2, 1, 0, 0, 0, -1, -2][d%8 + 1]))};

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

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 2*A[1] + 3*A[2];

Expansion of f(-q^3, -q^5)^4 / psi(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [ 2, 0, -2, 2, -2, 0, 2, -2, ...].
Moebius transform is period 8 sequence [ 2, 1, 0, 0, 0, -1, -2, 0, ...].
Convolution square of A093709.
a(2*n) = A244540(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 2*A053692(n). a(8*n + 7) = 0.

A244553 Expansion of phi(q^2) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 2, -1, 0, 2, 0, -1, 3, -4, 2, 2, 0, 0, 0, -1, 2, 1, 2, -4, 0, 2, 0, 2, 1, -4, 4, 0, 0, 0, 0, -1, 4, -2, 0, 1, 0, 2, 0, -4, 2, 0, 2, 2, 0, 0, 0, 2, 1, -5, 4, -4, 0, 4, 0, 0, 4, -4, 2, 0, 0, 0, 0, -1, 0, 4, 2, -2, 0, 0, 0, 1, 2, -4, 2, 2, 0, 0, 0, -4, 5

Offset: 1

Michael Somos, Jun 30 2014

sign

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

			G.f. = q - q^2 + 2*q^3 - q^4 + 2*q^6 - q^8 + 3*q^9 - 4*q^10 + 2*q^11 + ...

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

Cf. A033761, A113411, A112603, A244554.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] - A[3];

a[ n_] := If[ n < 1, 0, Sum[ {1, -2, 1, 0, -1, 2, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^2] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -2, 1, 0, -1, 2, -1][d%8 + 1]))};

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

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] - A[2];

Expansion of q * f(-q, -q^7)^2 * phi(q^2) / psi(-q) = q * f(-q, -q^7)^2 * chi(q^2)^2 / chi(-q) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [ -1, 2, 1, -2, 1, 2, -1, -2, ...].
Moebius transform is period 8 sequence [ 1, -2, 1, 0, -1, 2, -1, 0, ...].
a(2*n) = - A244554(n). a(2*n + 1) = A113411(n). a(8*n + 1) = A112603(n). a(8*n + 3) = 2 * A033761(n). a(8*n + 5) = a(8*n + 7) = 0.

cp's OEIS Frontend

A224326 Number of partitions of n into 3 distinct triangular numbers.

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