A053256 -id:A053256 - 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22.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

A259920 Expansion of phi(-x^5) * f(-x^5) / f(-x, -x^4) in powers of x where phi() and f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 5, 3, 5, 2, 6, 3, 6, 3, 7, 4, 7, 5, 9, 5, 9, 5, 11, 6, 11, 7, 14, 7, 15, 9, 17, 9, 17, 9, 21, 11, 21, 12, 25, 13, 25, 15, 29, 16, 31, 17, 35, 19, 37, 21, 42, 22, 44, 25, 49, 27, 52, 29, 58, 32, 61

Offset: 0

Michael Somos, Jul 08 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 + x + x^2 + x^3 + 2*x^4 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + x^11 + ...
G.f. = q^-1 + q^59 + q^119 + q^179 + 2*q^239 + q^359 + q^419 + 2*q^479 + q^539 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 8th equation.

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. A053256, A053266.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ -1, 0, 0, -1, 2, -1, 0, 0, -1, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];

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

Expansion of f(-x^5)^3 / (f(-x^10) * f(-x^2, -x^3)) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of phi(-x^5) * G(x) 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 10 sequence [ 1, 0, 0, 1, -2, 1, 0, 0, 1, -1, ...].
G.f.: (Sum_{k in Z} (-1)^k * x^(5*k^2)) / (Product_{k in Z} 1 - x^abs(5*k + 1)).

A306707 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 + j*x^j).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, -2, 2, 2, -1, -7, 5, 8, -4, -22, 23, 30, -32, -101, 102, 160, -81, -434, 288, 515, -249, -1393, 1542, 1756, -1950, -6694, 7040, 10192, -4724, -29594, 18851, 30282, -9298, -89627, 106068, 97255, -127909, -453650, 498939, 641627, -207845, -2026461

Offset: 0

Ilya Gutkovskiy, Mar 05 2019

sign

Cf. A053256, A318770.

nmax = 46; CoefficientList[Series[Sum[x^(k^2)/Product[(1 + j x^j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

A306708 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 + x^j)^j.

Original entry on oeis.org

1, 1, -1, 1, 0, 0, -2, 2, 1, 0, -4, 2, 0, 3, -3, 5, -6, -1, -7, 16, -3, 7, -19, 14, -14, 11, -23, 32, -16, 37, -38, 32, -59, 20, -47, 101, -40, 70, -122, 79, -87, 147, -108, 119, -270, 116, -199, 414, -49, 353, -481, 129, -687, 428, -376, 1003, -409, 759, -1114

Offset: 0

Ilya Gutkovskiy, Mar 05 2019

sign

Cf. A053256, A318771.

nmax = 58; CoefficientList[Series[Sum[x^(k^2)/Product[(1 + x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

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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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A259920 Expansion of phi(-x^5) * f(-x^5) / f(-x, -x^4) in powers of x where phi() and f() are Ramanujan theta functions.

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A306707 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 + j*x^j).

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A306708 Expansion of Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 + x^j)^j.

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