A053258 -id:A053258 - cp's OEIS frontend

A053257 Coefficients of the '5th-order' mock theta function f_1(q).

1, 0, 1, -1, 1, -1, 2, -2, 1, -1, 2, -2, 2, -2, 2, -3, 3, -2, 3, -4, 4, -4, 4, -5, 5, -4, 5, -6, 6, -6, 7, -8, 7, -7, 8, -9, 10, -9, 10, -12, 11, -11, 13, -14, 14, -15, 16, -17, 17, -16, 19, -21, 20, -21, 23, -25, 25, -25, 27, -29, 30, -30, 32, -35, 35, -35, 39, -41, 41, -43, 45, -49, 50, -49, 53, -57, 58, -59, 63, -67, 68

Offset: 0

Dean Hickerson, Dec 19 1999

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Other '5th-order' mock theta functions are at A053256, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

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

G.f.: f_1(q) = Sum_{n>=0} q^(n^2+n)/((1+q)(1+q^2)...(1+q^n)).
Consider partitions of n into parts differing by at least 2 and with smallest part at least 2. a(n) is the number of them with largest part even minus number with largest part odd.
a(n) ~ (-1)^n * sqrt(phi) * exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 15 2019

A053259 Coefficients of the '5th-order' mock theta function phi_1(q).

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 4, 2, 2, 4, 4, 3, 3, 4, 4, 4, 3, 4, 5, 4, 4, 5, 5, 4, 4, 5, 6, 5, 4, 6, 7, 5, 5, 6, 7, 6, 6, 7, 7, 7, 6, 8, 9, 7, 7, 9

Offset: 0

Dean Hickerson, Dec 19 1999

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22, 25.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053260, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Series[Sum[q^(n+1)^2 Product[1+q^(2k-1), {k, 1, n}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[x^((k+1)^2) * Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

G.f.: phi_1(q) = Sum_{n>=0} q^(n+1)^2 (1+q)(1+q^3)...(1+q^(2n-1)).
a(n) is the number of partitions of n into odd parts such that each occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive odd numbers occur.
a(n) ~ exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 3, 3, 3, 4, 4, 5, 5, 7, 9, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 19, 24, 23, 25, 27, 28, 31, 32, 33, 37, 40, 42, 44, 47, 52, 54, 59, 62, 67, 75, 75, 80, 87, 90, 95, 102, 109, 114, 119, 127, 134, 142, 150, 159, 171, 178, 187, 199, 211

Offset: 0

Vaclav Kotesovec, Sep 29 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A053258, A216222, A306734, A369557, A376542, A376581, A376621.

nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ c * A376621^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.390989767113799449629...
a(n) ~ c * A376542(n), where c = (108 + 12*sqrt(93))^(1/3)/3 - 4/(108 + 12*sqrt(93))^(1/3) = 1.364655607... is the real root of the equation c*(4 + c^2) = 8.
a(n) ~ c * A369557(n), where c = A347178 = -sinh(log((-3*sqrt(3) + sqrt(31))/2)/3) / sqrt(3) = 0.3411639019... is the real root of the equation 2*c*(1 + 4*c^2) = 1.
a(n) ~ A376631(n) * (A376621/A376660)^sqrt(n).

A376631 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j)).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 1, 1, 1, 1, 2, 0, 3, 0, 2, 1, 3, 1, 3, 1, 2, 3, 2, 3, 2, 4, 1, 5, 2, 5, 2, 6, 1, 7, 2, 7, 3, 6, 4, 7, 5, 6, 7, 6, 7, 7, 9, 5, 11, 5, 12, 6, 14, 5, 15, 6, 16, 7, 17, 7, 18, 9, 18, 11, 19, 12, 20, 14, 19, 17, 19, 19, 20, 23, 18, 27, 18, 29, 20, 32, 19

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A053258, A053261, A264905, A333179, A376580, A376632, A376660.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j)).
a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = 1/(2*sqrt(3 - 4*sinh(arcsinh(3^(3/2)/2) / 3) / sqrt(3))) = 0.39098976711379944962936707496887239986756106886318...
a(n) ~ A376580(n) * (A376660/A376621)^sqrt(n).

A376630 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j-1)).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 2, 2, 0, 1, 2, 2, 1, 1, 2, 3, 2, 1, 3, 2, 2, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 5, 5, 5, 4, 4, 7, 7, 5, 6, 8, 7, 7, 6, 8, 10, 8, 8, 10, 11, 9, 10, 12, 12, 11, 12, 14, 14, 13, 13, 16, 17, 15, 17, 18, 18, 19, 19, 20, 21, 22, 22, 24, 24, 25, 26, 27, 28, 29, 30

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A053258, A053261, A333179, A376629, A376660.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2)*Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*x^k]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[2*nmax]}]; Take[CoefficientList[s, x], nmax + 1]

G.f.: Sum_{k>=0} Product_{j=1..k} (x^j + x^(3*j-1)).

a(n) ~ c * A376660^sqrt(n) / sqrt(n), where c = sqrt(cosh(arccosh(sqrt(31)/2) / 3))/31^(1/4) = 0.456748282933947534736955792823221857...

A376628 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1)).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 10, 10, 12, 14, 15, 17, 19, 21, 23, 27, 29, 31, 37, 39, 43, 49, 52, 58, 64, 70, 76, 84, 92, 99, 111, 119, 129, 143, 153, 167, 183, 197, 213, 233, 251, 271, 295, 317, 343, 372, 400, 430, 466, 500, 538, 582, 622, 670

Offset: 0

Vaclav Kotesovec, Sep 30 2024

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003106, A053251, A053254, A053258, A053282, A333179.

nmax=100; CoefficientList[Series[Sum[x^(k*(k+1))/Product[1-x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} Product_{j=1..k} x^(2*j)/(1 - x^(2*j-1)).
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/2) * sqrt(n)).
Conjectural g.f.: (1 + q * nu(-q))/(1 + q) = 1 + Sum_{k >= 0} q^(k+2)*Product_{j = 1..k} 1 + q^(2*j+1), where nu(q) is the g.f. of A053254. - Peter Bala, Jan 03 2025

A376629 G.f.: Sum_{k>=0} x^(k(k+1)) Product_{j=1..k} (1 + x^(2*j-1)).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 2, 2, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 3, 3, 1, 3, 3, 3, 2, 2, 3, 3, 3, 2, 4, 3, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 4, 6, 6, 4, 6, 5, 7, 6, 5, 7, 7, 6, 6, 8, 7, 7, 7, 9, 8

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003106, A053251, A053258, A333179, A376630.

nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1))*Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

G.f.: Sum_{k>=0} Product_{j=1..k} (x^(2*j) + x^(4*j-1)).

a(n) ~ exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(n)).

A259551 Expansion of f(x^2, x^3) * f(-x^4, -x^6) / f(-x^2) in powers of x where f(,) is the Ramanujan general theta function.

Original entry on oeis.org

1, 0, 2, 1, 2, 1, 2, 1, 3, 2, 4, 4, 5, 4, 6, 5, 7, 6, 9, 8, 11, 11, 13, 13, 17, 15, 20, 19, 23, 23, 27, 27, 33, 33, 38, 39, 45, 45, 53, 54, 62, 63, 73, 74, 84, 86, 97, 100, 112, 115, 130, 134, 148, 154, 170, 176, 195, 202, 222, 232, 255, 264, 290, 301, 329

Offset: 0

Michael Somos, Jun 30 2015

nonn

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

Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

			G.f. = 1 + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 3*x^8 + 2*x^9 + ...
G.f. = 1/q + 2*q^239 + q^359 + 2*q^479 + q^599 + 2*q^719 + q^839 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 9th equation.

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 97, Equation (3.5)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053258, A053266, A255065, A259910.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, -2, -1, 1, 1, 1, -1, -2, 0, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[x^5] QPochhammer[ -x^2, x^5] QPochhammer[ -x^3, x^5] / (QPochhammer[ x^2, x^10]  QPochhammer[ x^8, x^10]), {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -2, -1, 1, 1, 1, -1, -2, 0][k%10 + 1]), n))};

Expansion of f(x^2, x^3) * G(x^2) in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function.
Euler transform of period 10 sequence [ 0, 2, 1, -1, -1, -1, 1, 2, 0, -1, ...].
G.f.: Product_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 3)) * (1 + x^(5*k - 2)) / ((1 - x^(10*k - 8)) * (1 - x^(10*k - 2))).
G.f.: (Sum_{k in Z} x^(k*(5*k + 1)/2)) / (Product_{k in Z} 1 - x^abs(10*k + 2)). - Michael Somos, Jul 09 2015
(-1)^n * A053258(n) + A053266(n) = a(n) unless n=0. Michael Somos, Jul 09 2015
A259910(n) = 2*A255065(n) + a(n). Michael Somos, Jul 09 2015

A260971 Expansion of phi_0(-q) in powers of q where phi_0() is a 5th-order mock theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 06 2015

sign

			G.f. = 1 - x + x^2 + x^4 - x^5 - x^7 + x^8 - x^9 + x^10 + x^12 - x^13 + ...
G.f. = q^-1 - q^119 + q^239 + q^479 - q^599 - q^839 + q^959 - q^1079 + ...

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

Cf. A053258, A053262, A053264.

a[n_]:= SeriesCoefficient[Sum[(-x)^( k^2)*Product[1 - x^(2*j - 1), {j, 1, k}], {k, 0, Sqrt[n]}], {x, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Aug 01 2018 *)

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

G.f.: Sum_{k >= 0}  (-x)^n^2 * (1 - x) * (1 - x^3) * ... * (1 - x^(2*k-1)).
a(n) = (-1)^n * A053258(n) = 2 * A053264(n) - A053262(n).
a(n) ~ (-1)^n * sqrt(phi) * exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 15 2019

cp's OEIS Frontend

A053257 Coefficients of the '5th-order' mock theta function f_1(q).

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A053259 Coefficients of the '5th-order' mock theta function phi_1(q).

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A376580 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j-1))^2.

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A376631 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j)).

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A376630 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} (1 + x^(2*j-1)).

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A376628 G.f.: Sum_{k>=0} x^(k*(k+1)) / Product_{j=1..k} (1 - x^(2*j-1)).

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A376629 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^(2*j-1)).

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A259551 Expansion of f(x^2, x^3) * f(-x^4, -x^6) / f(-x^2) in powers of x where f(,) is the Ramanujan general theta function.

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A260971 Expansion of phi_0(-q) in powers of q where phi_0() is a 5th-order mock theta function.

A376631 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j)).

A376630 G.f.: Sum_{k>=0} x^(k(k+1)/2) Product_{j=1..k} (1 + x^(2*j-1)).

A376628 G.f.: Sum_{k>=0} x^(k(k+1)) / Product_{j=1..k} (1 - x^(2j-1)).

A376629 G.f.: Sum_{k>=0} x^(k(k+1)) Product_{j=1..k} (1 + x^(2*j-1)).