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

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

Author

Michael Somos, Feb 18 2017

Keywords

• nonn

Comments

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.

Examples

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

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20

Links

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

Crossrefs

Essentially the same as A053267.

Programs

Mathematica

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}]]];

PARI

{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))};

PARI

{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))};

Formula

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