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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A106459 Expansion of f(-x, -x^3) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, May 02 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is a expansion of Ramanujan's general theta function in powers of x because |a(n)| = A010054(n) is also the characteristic function of generalized hexagonal numbers. - Omar E. Pol, Jun 13 2012
Number 4 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
Also the number of partitions of n into an even number of parts, where each part occurs at most 3 times, minus the number of partitions of n into an odd number of parts, where each part occurs at most 3 times. - Jeremy Lovejoy, Aug 04 2020

Examples

			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. = q - q^9 - q^25 + q^49 + q^81 - q^121 - q^169 + q^225 + q^289 - q^361 - ...

References

  • D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 53, Exer. 2.2.10

Links

Crossrefs

Cf. A006950, A010054, A192540.

Programs

  • Mathematica
    a[ n_] := If[ SquaresR[ 1, 8 n + 1] == 2, (-1)^Quotient[ Sqrt[8 n + 1] + 1, 4], 0]; (* Michael Somos, Nov 18 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q] / (2^(1/2) q^(1/4)), {q, 0, 2 n}]; (* Michael Somos, Nov 18 2011 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) / eta(x^2 + A), n))}
  • PARI
    {a(n) = my(x); if( issquare( 8*n + 1, &x), kronecker( 2, x))};

Formula

Expansion of psi(-x) = f(x^6, x^10) - x * f(x^2, x^14) in powers of x where psi() is a Ramanujan theta function, and f(,) is Ramanujan's general theta function.
Expansion of q^(-1/8) * eta(q) * eta(q^4) / eta(q^2) in powers of q.
Euler transform of period 4 sequence [ -1, 0, -1, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 4 (t/i)^(1/2) f(t) where q = exp(2 Pi i t).
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) = u1^4*u6^4 + u1^3*u2*u3^3*u6 + 2*u1*u2^3*u3*u6^3 - u2^4*u3^4.
a(n) = b(8*n + 1) where b() is multiplicative with b(p^e) = Kronecker(2, p)^(e/2) if e even, b(p^e) = 0 if e odd.
G.f.: Product_{k>0} (1 - x^k) * (1 + x^(2*k)) = Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)).
G.f.: Product_{k>0} (1 - x^(2*k)) / (1 + x^(2*k - 1)) = Product_{k>0} (1 - x^(4*k)) * (1 - x^(2*k - 1)).
G.f.: Sum_{k>=0} a(k) * x^(8*k + 1) = Sum_{k in Z} (-1)^k * x^((4*k + 1)^2).
G.f.: Sum_{k>=0} (-x)^(k*(k + 1)/2) = Sum_{k in Z} x^(8*k^2 + 2*k) - x^(8*k^2 + 6*k + 1).
G.f. A(x) satisfies: x / A(F(x)) = F(x) = g.f. of A192540.
Convolution inverse of A006950.
|a(n)| = A010054(n) the characteristic function of triangular numbers.
G.f.: 1 + (-x)*(1 + (-x)^2*(1 + (-x)^3*(1 + ...))). - Michael Somos, Mar 03 2014

A296045 a(n) = [x^n] Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^n.

Original entry on oeis.org

1, 1, 3, 13, 55, 231, 981, 4222, 18351, 80320, 353453, 1562364, 6932185, 30856541, 137725710, 616190583, 2762605791, 12408541299, 55825435656, 251523510045, 1134741006825, 5125453110196, 23175983361270, 104899547541255, 475228898015025, 2154737528486881, 9777332125043577
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2017

Keywords

Links

Crossrefs

Cf. A006950, A106337, A192540, A273225, A273226, A273228, A295832, A296043, A296044.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^(4 k)))^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[(2 (-x)^(1/8)/EllipticTheta[2, 0, Sqrt[-x]])^n, {x, 0, n}], {n, 0, 26}]
Table[(-1)^n * 2^n * SeriesCoefficient[1/(QPochhammer[-1, x]*QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 07 2020 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, 4/Sqrt[Pi*(77/2 - 4*s*(-r*s)^(7/8) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[-r*s]])]} /. FindRoot[{s == (2*(-r*s)^(1/8))/EllipticTheta[2, 0, Sqrt[-r*s]], 7*I*r + 2*(-r*s)^(7/8)*Sqrt[r*s] * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[-r*s]] == 0}, {r, 1/5}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

Formula

a(n) = [x^n] Product_{k>=1} ((1 + x^k)/(1 - x^(4*k)))^n.
a(n) ~ c * d^n / sqrt(n), where d = 4.62579056836776492108784045382518984897... (see A192540) and c = 0.255113338880004277664416308115912337... - Vaclav Kotesovec, Dec 05 2017

A259938 Expansion of the series reversion of Sum_{n>=1} x^(n^2).

Original entry on oeis.org

0, 1, 0, 0, -1, 0, 0, 4, 0, -1, -22, 0, 13, 140, 0, -136, -970, 9, 1330, 7104, -231, -12650, -54096, 3900, 118780, 423890, -54810, -1108380, -3393696, 695640, 10311840, 27615648, -8282604, -95810606, -227480848, 94449456, 889817328, 1890685212, -1044402840, -8263944216, -15811484852
Offset: 0

Views

Author

Vladimir Reshetnikov, Jul 09 2015

Keywords

Comments

x + x^4 + x^9 + x^16 + x^25 + ... is the expansion of (theta_3(0, x) - 1)/2, where theta_3 is the Jacobi theta function.

Links

Crossrefs

Cf. A049140, A192540.

Programs

  • Mathematica
    InverseSeries[(EllipticTheta[3, 0, x] - 1)/2 + O[x]^30][[3]]
  • PARI
    Vec( serreverse( sum(i=1,32,x^i^2) + O(x^33^2) ) ); \\ Max Alekseyev, Jul 06 2021

Formula

For n>1, a(n) = Sum_{j2,j3,...} (-1)^(j2+j3+...) * (n-1+j2+j3+...)! / (j2!*j3!*...) / n!, where the sum is taken over all nonnegative integers j2, j3, ... such that (2^2-1)*j2 + (3^2-1)*j3 + ... = n-1. - Max Alekseyev, Jul 06 2021
Showing 1-3 of 3 results.

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