A053267 -id:A053267 - 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 22, 25.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

A259771 Expansion of x * psi(x^5) * f(-x^10) / f(-x^2,-x^8) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 3, 4, 4, 5, 4, 6, 5, 7, 7, 9, 8, 10, 10, 12, 12, 15, 14, 18, 17, 20, 20, 24, 24, 28, 28, 33, 33, 38, 38, 44, 45, 50, 52, 59, 60, 68, 69, 78, 80, 89, 92, 102, 105, 116, 120, 133, 137, 151, 156, 171, 178, 194, 201

Offset: 1

Michael Somos, Jul 04 2015

nonn

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

The g.f. for this sequence is the last term of the 14th equation on page 20 of Ramanujan 1988.

			G.f. = x + x^3 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + x^10 + 2*x^11 + x^12 + ...
G.f. = q^49 + q^289 + q^529 + q^649 + q^769 + q^889 + 2*q^1009 + q^1129 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 20

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053265, A053267.

a[ n_] := SeriesCoefficient[ x Product[ (1 - x^k)^{ 0, -1, 0, 0, -1, 0, 0, -1, 0, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}];
QP:= QPochhammer; a[n_]:= SeriesCoefficient[ x*QP[x^10]/(QP[x^5, x^10]* QP[x^2, x^10]*QP[x^8, x^10]), {x, 0, n}]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Mar 16 2018 *)

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

Euler transform of period 10 sequence [ 0, 1, 0, 0, 1, 0, 0, 1, 0, -1, ...].

a(n) = A053265(n-1) - A053267(n).

A282537 Coefficients of the '5th-order' mock theta function Psi(q) with a(0)=1.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 4, 4, 5, 5, 7, 6, 8, 8, 9, 9, 12, 11, 14, 14, 16, 16, 20, 19, 23, 24, 27, 27, 32, 32, 37, 38, 43, 44, 51, 51, 58, 61, 67, 69, 78, 80, 89, 93, 102, 106, 118, 121, 134, 140, 153, 159, 175, 181, 198, 207, 224, 234

Offset: 0

Michael Somos, Feb 18 2017

nonn

In Ramanujan's lost notebook the generating function is denoted by psi(q) on pages 18 and 20, however on page 20 there is a "-1" first term.

			G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + 2*x^11 + ...

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

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

Essentially the same as A053267.

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(5 k^2) / (QPochhammer[ x^2, x^5, k + 1] QPochhammer[ x^3, x^5, k]) // FunctionExpand, {k, 0, Sqrt[n/5]}], {x, 0, n}]];
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24(n + 2)/5]}, SeriesCoefficient[ Sum[ -(-1)^k x^(5 k (3 k + 1)/2 - 2) / (1 - x^(5 k - 2)), {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+2, 1 - if( i%5==2 || i%5==3, 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+2)\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, -(-1)^k * x^(5*k*(3*k + 1)/2 - 2) / (1 - x^(5*k - 2)), A) / eta(x^5 + A), n))};

G.f.: Sum_{k>=0} x^(5*k^2) / ((1 - x^2) * (1 - x^3) * (1 - x^7) * (1 - x^8)...(1 - x^(5*k+2))).
G.f.: ( Sum_{k in Z} -(-1)^k * x^(5*k*(3*k + 1)/2 - 2) / (1 - x^(5*k - 2)) ) / ( Sum_{k in Z} (-1)^k * x^(5*k*(3*k - 1)) ).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(3/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

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

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A259771 Expansion of x * psi(x^5) * f(-x^10) / f(-x^2,-x^8) in powers of x where psi(), f() are Ramanujan theta functions.

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A282537 Coefficients of the '5th-order' mock theta function Psi(q) with a(0)=1.

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