A053262 -id:A053262 - cp's OEIS frontend

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

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22, 25.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

A259910 Expansion of f(-x^2, -x^3)^3 / f(-x)^2 in powers of x where f(,) is the Ramanujan general theta function.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 4, 3, 3, 4, 6, 6, 7, 8, 8, 9, 11, 12, 13, 14, 17, 19, 21, 21, 25, 27, 30, 31, 35, 39, 43, 47, 51, 55, 60, 65, 71, 77, 83, 88, 98, 105, 115, 122, 132, 142, 155, 164, 178, 191, 206, 220, 236, 252, 272, 290, 311, 332, 356, 378, 407, 434, 464

Offset: 0

Michael Somos, Jul 07 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^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 3*x^7 + 3*x^8 + ...
G.f. = 1/q + 2*q^119 + 2*q^239 + q^359 + 2*q^479 + 3*q^599 + 4*q^719 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 6th equation.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053262, A053266.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -2, 1, 1, -2, 1}[[
Mod[k, 5, 1]]], {k, n}], {x, 0, n}];
nmax = 100; CoefficientList[Series[Product[(1-x^k)/((1 - x^(5*k-1))*(1 - x^(5*k-4)))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 17 2016 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -2, 1, 1, -2][k%5+1]), n))};

Expansion of f(-x) * (f(-x^5) / f(-x, -x^4))^3 in powers of x where f(,) is the Ramanujan general theta function.
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. - Michael Somos, Jul 09 2015
Euler transform of period 5 sequence [ 2, -1, -1, 2, -1, ...].
G.f.: (Sum_{k in Z} (-1)^k * x^(k*(5*k + 1)/2)) / (Product_{k in Z} 1 - x^abs(5*k + 1))^2.
a(n) = 3 * A053266(n) - A053262(n) unless n=0.
a(n) ~ sqrt(5+2*sqrt(5)) * exp(sqrt(2*n/15)*Pi)/ (5*sqrt(2*n)). - Vaclav Kotesovec, Dec 17 2016

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

A279135 Coefficients of the '5th-order' mock theta function Phi(q) with a(0)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 8, 8, 9, 10, 12, 12, 14, 15, 17, 18, 20, 21, 25, 26, 29, 31, 35, 36, 41, 43, 48, 51, 56, 59, 66, 70, 76, 81, 89, 94, 103, 109, 119, 126, 137, 144, 158, 167, 180, 191, 207, 218, 236, 250, 269, 285, 306, 323

Offset: 0

Michael Somos, Dec 06 2016

nonn

In Ramanujan's lost notebook the generating function is denoted by phi(q) on pages 18 and 20, however on page 20 there is a minus one first term.

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A053262. Essentially the same as A053266.

Cf. A259910.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(5 k^2) / (QPochhammer[ x, x^5, k + 1] QPochhammer[ x^4, x^5, k]) // FunctionExpand, {k, 0, Sqrt[n/5]}], {x, 0, n}]];
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24 n/5]}, SeriesCoefficient[ Sum[ (-1)^k x^(5 k (3 k + 1)/2) / (1 - x^(5 k + 1)), {k, Quotient[m + 1, -6], Quotient[m - 1, 6]}] / QPochhammer[ x^5], {x, 0, n}]]];

{a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n\5), x^(5*k^2) / prod(i=1, 5*k+1, 1 - if( i%5==1 || i%5==4, x^i), 1 + x * O(x^(n - 5*k^2)))), n))};

{a(n) = my(A, m); if( n<0, 0, m = sqrtint(1 + 24*n\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, (-1)^k * x^(5*k*(3*k + 1)/2) / (1 - x^(5*k + 1)), A) / eta(x^5 + A), n))};

G.f.: Sum_{k>=0} x^(5*k^2) / ((1 - x) * (1 - x^4) * (1 - x^6) * (1 - x^9)...(1 - x^(5*k+1))).
3*a(n) = A053262(n) + A259910(n) unless n=0. [Ramanujan, p. 23, equation 6]
a(n) ~ sqrt(phi/2) * exp(Pi*sqrt(2*n/15)) / (5^(3/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

cp's OEIS Frontend

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

A259910 Expansion of f(-x^2, -x^3)^3 / f(-x)^2 in powers of x where f(,) is the Ramanujan general theta function.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A279135 Coefficients of the '5th-order' mock theta function Phi(q) with a(0)=1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula