cp's OEIS Frontend

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.

Showing 1-10 of 11 results. Next

A053269 Coefficients of the '6th-order' mock theta function psi(q).

Original entry on oeis.org

0, 1, -1, 1, -2, 3, -2, 2, -4, 5, -5, 5, -7, 9, -8, 9, -12, 14, -15, 16, -20, 23, -23, 25, -31, 36, -37, 40, -47, 54, -56, 60, -71, 79, -84, 91, -103, 115, -121, 131, -149, 164, -174, 188, -210, 232, -245, 264, -294, 321, -343, 368, -406, 443, -470, 505, -554, 602, -641, 687, -751, 813, -863, 925
Offset: 0

Views

Author

Dean Hickerson, Dec 19 1999

Keywords

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 4, 13

Links

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053270, A053271, A053272, A053273, A053274.
Cf. A262614, A263041.

Programs

  • Mathematica
    Series[Sum[(-1)^n q^(n+1)^2 Product[1-q^k, {k, 1, 2n-1, 2}]/Product[1+ q^k, {k, 1, 2n+1}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[(-1)^k * x^((k+1)^2) * Product[1-x^j, {j, 1, 2*k-1, 2}]/Product[1+ x^j, {j, 1, 2*k+1}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)

Formula

G.f.: psi(q) = Sum_{n >= 0} (-1)^n q^(n+1)^2 (1-q)*(1-q^3)...(1-q^(2n-1)) /((1+q)*(1+q^2)...(1+q^(2n+1))).
a(3*n + 1) = A262614(n). a(3*n + 2) = - A263041(n). - Michael Somos, Apr 17 2016
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019

A053270 Coefficients of the '6th-order' mock theta function rho(q).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 14, 18, 24, 30, 38, 47, 58, 72, 88, 108, 130, 156, 188, 225, 268, 318, 376, 444, 522, 612, 716, 834, 972, 1129, 1308, 1512, 1744, 2010, 2310, 2652, 3038, 3474, 3968, 4524, 5152, 5857, 6650, 7542, 8540, 9660, 10912, 12312, 13878
Offset: 0

Views

Author

Dean Hickerson, Dec 19 1999

Keywords

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 3, 13

Links

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053269, A053271, A053272, A053273, A053274.

Programs

  • Mathematica
    Series[Sum[q^(n(n+1)/2) Product[1+q^k, {k, 1, n}]/Product[1-q^k, {k, 1, 2n+1, 2}], {n, 0, 13}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}]/Product[1-x^j, {j, 1, 2*k+1, 2}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Formula

G.f.: rho(q) = Sum_{n >= 0} ( q^(n(n+1)/2) *(1+q)*(1+q^2)...(1+q^n)/((1-q)*(1-q^3)...(1-q^(2n+1))) ).
a(n) ~ exp(Pi*sqrt(n/3)) / (2*sqrt(3*n)). - Vaclav Kotesovec, Jun 12 2019

A053274 Coefficients of the '6th-order' mock theta function gamma(q).

Original entry on oeis.org

1, 1, -1, 0, 2, -2, -1, 3, -2, 0, 3, -4, -1, 5, -3, -1, 6, -6, -2, 7, -6, 0, 9, -8, -3, 11, -9, -2, 13, -13, -3, 17, -12, -3, 19, -18, -5, 22, -19, -3, 27, -24, -7, 33, -26, -5, 36, -34, -9, 44, -35, -9, 51, -45, -11, 58, -49, -9, 68, -59, -16, 78, -65, -15, 88, -79, -19, 104, -84, -19, 117, -102, -26, 133, -112, -24, 152, -131
Offset: 0

Views

Author

Dean Hickerson, Dec 19 1999

Keywords

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17

Links

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053273.
Cf. A328988, A328989.

Programs

  • Mathematica
    Series[Sum[q^n^2/Product[1+q^k+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
  • PARI
    a(n) = polcoeff(sum(k=0, 50, q^(k^2)/prod(j=1, k, 1+q^j+q^(2*j)), q*O(q^n)), n);
for(n=0,50, print1(a(n), ", ")) \\ G. C. Greubel, May 18 2018
  • PARI
    my(N=80, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1+x^k)*(1-x^k)^2/(1+x^k+x^(2*k)))) \\ Seiichi Manyama, May 23 2023

Formula

G.f.: gamma(q) = Sum_{n >= 0} q^n^2/((1+q+q^2)(1+q^2+q^4)...(1+q^n+q^(2n))).
From Seiichi Manyama, May 23 2023: (Start)
a(n) = A328988(n) - A328989(n) for n > 0.
G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1+x^k) * (1-x^k)^2 / (1+x^k+x^(2*k)). (End)

A053271 Coefficients of the '6th-order' mock theta function sigma(q).

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 5, 7, 8, 11, 14, 17, 22, 28, 33, 41, 51, 60, 74, 89, 105, 127, 151, 177, 210, 248, 289, 340, 398, 461, 537, 624, 719, 832, 960, 1101, 1267, 1453, 1660, 1899, 2167, 2465, 2807, 3190, 3614, 4097, 4638, 5237, 5915, 6671, 7507, 8450, 9498
Offset: 0

Views

Author

Dean Hickerson, Dec 19 1999

Keywords

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13.

Links

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053272, A053273, A053274.

Programs

  • Mathematica
    Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}]/Product[1-q^k, {k, 1, 2n+1, 2}], {n, 0, 12}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^((k+1)*(k+2)/2) * Product[1+x^j, {j, 1, k}]/Product[1-x^j, {j, 1, 2*k+1, 2}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Formula

G.f.: sigma(q) = Sum_{n >= 0} q^((n+1)(n+2)/2) (1+q)(1+q^2)...(1+q^n)/((1-q)(1-q^3)...(1-q^(2n+1))).
a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(3*n)). - Vaclav Kotesovec, Jun 12 2019

A053272 Coefficients of the '6th-order' mock theta function lambda(q).

Original entry on oeis.org

1, -1, 3, -5, 6, -7, 11, -16, 18, -21, 30, -40, 47, -56, 72, -92, 108, -125, 156, -193, 225, -263, 318, -383, 444, -513, 612, -724, 834, -963, 1129, -1320, 1512, -1730, 2010, -2325, 2652, -3022, 3474, -3988, 4524, -5129, 5857, -6673, 7542, -8515, 9660, -10943, 12312, -13842
Offset: 0

Views

Author

Dean Hickerson, Dec 19 1999

Keywords

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13

Links

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053273, A053274.

Programs

  • Mathematica
    Series[Sum[(-q)^n Product[1-q^k, {k, 1, 2n-1, 2}]/Product[1+q^k, {k, 1, n}], {n, 0, 100}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[(-x)^k * Product[1-x^j, {j, 1, 2*k-1, 2}] / Product[1+x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)

Formula

G.f.: lambda(q) = Sum_{n >= 0} (-q)^n (1-q)(1-q^3)...(1-q^(2n-1))/((1+q)(1+q^2)...(1+q^n)).
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019

A053273 Coefficients of the '6th-order' mock theta function 2 mu(q).

Original entry on oeis.org

1, 2, -3, 4, -4, 6, -11, 14, -15, 22, -31, 34, -41, 56, -69, 82, -98, 120, -152, 178, -204, 254, -308, 354, -415, 496, -587, 680, -785, 922, -1084, 1248, -1427, 1664, -1935, 2202, -2517, 2906, -3336, 3798, -4315, 4930, -5636, 6380, -7202, 8194, -9305, 10474, -11801, 13342, -15050
Offset: 0

Views

Author

Dean Hickerson, Dec 19 1999

Keywords

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 13

Links

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053274.

Programs

  • Mathematica
    Series[1+Sum[(-1)^n q^(n+1) (1+q^n) Product[1-q^k, {k, 1, 2n-1, 2}]/Product[1+q^k, {k, 1, n+1}], {n, 0, 99}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[1 + Sum[(-1)^k * x^(k+1) * (1+x^k) * Product[1-x^j, {j, 1, 2*k-1, 2}] / Product[1+x^j, {j, 1, k+1}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)

Formula

G.f.: 2 mu(q) = 1 + Sum_{n >= 0} (-1)^n q^(n+1) (1+q^n) (1-q)(1-q^3)...(1-q^(2n-1))/((1+q)(1+q^2)...(1+q^(n+1))).
a(n) ~ -(-1)^n * exp(Pi*sqrt(n/3)) / (2*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019

A153251 Coefficients of the sixth-order mock theta function phi_{-}(q).

Original entry on oeis.org

0, 1, 3, 5, 8, 14, 22, 33, 51, 74, 105, 151, 210, 289, 398, 537, 719, 960, 1267, 1660, 2167, 2807, 3614, 4638, 5915, 7507, 9498, 11957, 14994, 18744, 23337, 28959, 35834, 44192, 54338, 66643, 81499, 99407, 120969, 146836, 177820
Offset: 0

Views

Author

Jeremy Lovejoy, Dec 21 2008

Keywords

Links

Crossrefs

Cf. A153252.
Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053273, A053274.

Programs

  • PARI
    lista(nn) = q = qq + O(qq^nn); gf = sum(n = 1, nn, q^n * prod(k = 1, 2*n-1, 1 + q^k) / prod(k = 1, n, 1 - q^(2*k-1))); concat(0, Vec(gf)) \\ Michel Marcus, Jun 18 2013

Formula

G.f.: Sum_{n >= 1} q^n (1+q)(1+q^2)...(1+q^(2n-1))/((1-q)(1-q^3)...(1-q^(2n-1))).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*sqrt(3*n)). - Vaclav Kotesovec, Jun 13 2019

A153252 Coefficients of the sixth-order mock theta function psi_{-}(q).

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 19, 29, 44, 65, 94, 134, 188, 261, 358, 486, 654, 872, 1155, 1519, 1984, 2576, 3325, 4270, 5456, 6939, 8786, 11077, 13912, 17406, 21700, 26961, 33388, 41221, 50739, 62278, 76232, 93067, 113336, 137684, 166873
Offset: 0

Views

Author

Jeremy Lovejoy, Dec 21 2008

Keywords

Links

Crossrefs

Cf. A153251.
Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053273, A053274.

Programs

  • PARI
    lista(nn) =  q = qq + O(qq^nn); gf = sum(n = 1, nn, q^n * prod(k = 1, 2*n-2, 1 + q^k) / prod(k = 1, n, 1 - q^(2*k-1))); concat(0, Vec(gf)) \\Michel Marcus, Jun 18 2013

Formula

G.f.: Sum_{n >= 1} q^n(1+q)(1+q^2)...(1+q^(2n-2))/((1-q)(1-q^3)...(1-q^(2n-1))).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*sqrt(3*n)). - Vaclav Kotesovec, Jun 13 2019

A261454 Expansion of a(x^2) / f(-x) in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 8, 9, 17, 25, 47, 63, 106, 144, 216, 296, 425, 569, 807, 1064, 1449, 1905, 2551, 3304, 4353, 5592, 7254, 9247, 11859, 14978, 19038, 23872, 30034, 37433, 46734, 57854, 71739, 88305, 108766, 133191, 163099, 198697, 242069, 293535, 355788, 429609, 518396
Offset: 0

Views

Author

Michael Somos, Nov 18 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + 8*x^2 + 9*x^3 + 17*x^4 + 25*x^5 + 47*x^6 + 63*x^7 + ...
G.f. = 1/q + q^23 + 8*q^47 + 9*q^71 + 17*q^95 + 25*q^119 + 47*q^143 + ...

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 1st equation.

Links

Crossrefs

Cf. A053268, A153251.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 + 9 x^2 QPochhammer[ x^18]^3) / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}];
nmax = 50; CoefficientList[Series[Product[(1 + x^k)^3*(1 - x^k)^2/(1 - x^(6*k)), {k, 1, nmax}] + 9*x^2*Product[(1 - x^(18*k))^3/((1 - x^k)*(1 - x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 + 9 * x^2 * eta(x^18 + A)^3) / (eta(x + A) * eta(x^6 + A)), n))};

Formula

Expansion of q^(1/24) * (eta(q^2)^3 + 9 * eta(q^18)^3) / (eta(q) * eta(q^6)) in powers of q.
Expansion of phi(x) + 2*phi_{-}(x) in powers of x where phi() and phi_{-}() are 6th-order mock theta functions. [Ramanujan]
a(n) = A053268(n) + 2*A153251(n). [Ramanujan]
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(3/2)*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019

A264136 Expansion of f(-q) * phi(q) in powers of q where f() is a Ramanujan theta function and phi() is a 6th-order mock theta function.

Original entry on oeis.org

1, -2, 2, -2, 0, -2, 4, 0, 2, -2, 2, -4, -2, 0, 6, -2, 0, -4, 4, 0, -2, -2, 2, -4, 2, 2, 8, -2, -2, -4, 2, 0, 2, -2, 0, -4, -2, 0, 8, -2, 0, -4, 6, 0, -2, 0, 0, -4, 0, -2, 6, -2, -2, -4, 4, 2, 6, 0, 0, -4, -2, 0, 8, -4, 0, -2, 2, 0, -2, -4, -2, -4, 4, 0, 8, 2
Offset: 0

Views

Author

Michael Somos, Nov 03 2015

Keywords

Examples

			G.f. = 1 - 2*x + 2*x^2 - 2*x^3 - 2*x^5 + 4*x^6 + 2*x^8 - 2*x^9 + 2*x^10 - 4*x^11 + ...

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 2, 2nd equation.

Links

Crossrefs

Cf. A010815, A053268.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, SeriesCoefficient[ QPochhammer[ x] Sum[ (-1)^k x^k^2 QPochhammer[ x, x^2, k] / QPochhammer[ -x, x, 2*k], {k, 0, Sqrt@n}], {x, 0, n}]];
nmax = 122; CoefficientList[Series[QPochhammer[q]*Sum[(-1)^n*q^n^2*Product[1 - q^k, {k, 1, 2*n - 1, 2}] / Product[1 + q^k, {k, 1, 2*n}], {n, 0, Floor[Sqrt[nmax]]}], {q, 0, nmax}], q] (* G. C. Greubel, Mar 18 2018, fixed by Vaclav Kotesovec, Jun 15 2019 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n)) * sum(k=0, sqrtint(n), (-1)^k * x^k^2 * prod(i=1, k, 1 - x^(2*i - 1), 1 + x * O(x^(n - k^2))) / prod(i=1, 2*k, 1 + x^i, 1 + x * O(x^(n - k^2))) ), n))};

Formula

Convolution of A010815 and A053268.
G.f.: Sum_{k in Z} x^(6*k^2 + k) / (1 - x^k + x^(2*k)) - 2 * Sum_{k in Z} x^(6*k^2 - 2*k) / (1 + x^(3*k - 1)).
Showing 1-10 of 11 results. Next

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