A053274 - OEIS

A053274

Coefficients of the '6th-order' mock theta function gamma(q).

1, 1, -1, 0, 2, -2, -1, 3, -2, 0, 3, -4, -1, 5, -3, -1, 6, -6, -2, 7, -6, 0, 9, -8, -3, 11, -9, -2, 13, -13, -3, 17, -12, -3, 19, -18, -5, 22, -19, -3, 27, -24, -7, 33, -26, -5, 36, -34, -9, 44, -35, -9, 51, -45, -11, 58, -49, -9, 68, -59, -16, 78, -65, -15, 88, -79, -19, 104, -84, -19, 117, -102, -26, 133, -112, -24, 152, -131 (list; graph; refs; listen; history; text; internal format)

	OFFSET	`0,5`
	REFERENCES	`Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17`
	LINKS	`G. C. Greubel, Table of n, a(n) for n = 0..1000` `George E. Andrews and Dean Hickerson, Ramanujan's "lost" notebook VII: The sixth order mock theta functions, Advances in Mathematics, 89 (1991) 60-105.`
	FORMULA	`G.f.: gamma(q) = Sum_{n >= 0} q^n^2/((1+q+q^2)(1+q^2+q^4)...(1+q^n+q^(2n))).` `From Seiichi Manyama, May 23 2023: (Start)` `a(n) = A328988(n) - A328989(n) for n > 0.` `G.f.: 1 + (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k(3k-1)/2) * (1+x^k) * (1-x^k)^2 / (1+x^k+x^(2*k)). (End)`
	MATHEMATICA	`Series[Sum[q^n^2/Product[1+q^k+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]`
	PROG	`(PARI) a(n) = polcoeff(sum(k=0, 50, q^(k^2)/prod(j=1, k, 1+q^j+q^(2j)), qO(q^n)), n);` `for(n=0, 50, print1(a(n), ", ")) \\ G. C. Greubel, May 18 2018` `(PARI) my(N=80, x='x+O('x^N)); Vec(1+1/prod(k=1, N, 1-x^k)sum(k=1, N, (-1)^(k-1)x^(k(3k-1)/2)(1+x^k)(1-x^k)^2/(1+x^k+x^(2*k)))) \\ Seiichi Manyama, May 23 2023`
	CROSSREFS	`Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053273.` `Cf. A328988, A328989.` `Sequence in context: A057036 A069004 A182490 * A243926 A281013 A190683` `Adjacent sequences: A053271 A053272 A053273 * A053275 A053276 A053277`
	KEYWORD	`sign,easy`
	AUTHOR	`Dean Hickerson, Dec 19 1999`
	STATUS	`approved`