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A279135 Coefficients of the '5th-order' mock theta function Phi(q) with a(0)=1. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

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.

REFERENCES

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20, 23

LINKS

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

FORMULA

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

EXAMPLE

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 + ...

MATHEMATICA

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}]]];

PROG

(PARI) {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))};

(PARI) {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))};

CROSSREFS

Cf. A053262. Essentially the same as A053266.

Cf. A259910.

Sequence in context: A324918 A068980 A280950 * A053266 A112217 A172033

Adjacent sequences:  A279132 A279133 A279134 * A279136 A279137 A279138

KEYWORD

nonn

AUTHOR

Michael Somos, Dec 06 2016

STATUS

approved

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Last modified January 27 17:36 EST 2022. Contains 350611 sequences. (Running on oeis4.)