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Search: id:A053250
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| A053250 |
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Coefficients of the '3rd order' mock theta function phi(q) |
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+0
12
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| 1, 1, 0, -1, 1, 1, -1, -1, 0, 2, 0, -2, 1, 1, -1, -2, 1, 3, -1, -2, 1, 2, -2, -3, 1, 4, 0, -4, 2, 3, -2, -4, 1, 5, -2, -5, 3, 5, -3, -5, 2, 7, -2, -7, 3, 6, -4, -8, 3, 9, -2, -9, 5, 9, -5, -10, 3, 12, -4, -12, 5, 11, -6, -13, 6, 16, -6, -15, 7, 15, -8, -17, 7, 19, -6, -20, 9, 19, -10, -22, 8, 25, -9, -25, 12, 25, -12, -27, 11, 31
(list; graph; listen)
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OFFSET
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0,10
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REFERENCES
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Leila A. Dragonette, Some asymptotic formulae for the mock theta functions of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.12), p. 58, Eq. (26.56).
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FORMULA
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Consider partitions of n into distinct odd parts. a(n) = number of them for which the largest part minus twice the number of parts is == 3 (mod 4) minus the number for which it is == 1 (mod 4).
a(n) = (-1)^n*(A027358(n)-A027357(n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 12 2006
G.f.: 1+ Sum_{k>0} x^k^2/((1+x^2)(1+x^4)...(1+x^(2k))).
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MAPLE
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f:=n->q^(n^2)/mul((1+q^(2*i)), i=1..n); add(f(n), n=0..10);
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MATHEMATICA
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Series[Sum[q^n^2/Product[1+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]
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PROGRAM
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(PARI) {a(n)= local(t); if(n<0, 0, t=1+O(x^n); polcoeff( sum(k=1, sqrtint(n), t*= x^(2*k-1)/ (1+x^(2*k)) +O(x^(n-(k-1)^2+1)), 1), n))} /* Michael Somos Jul 16 2007 */
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CROSSREFS
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Other '3rd order' mock theta functions are at A000025, A053251, A053252, A053253, A053254, A053255.
Sequence in context: A068320 A111330 A117447 this_sequence A160813 A116664 A024161
Adjacent sequences: A053247 A053248 A053249 this_sequence A053251 A053252 A053253
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KEYWORD
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sign,easy
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AUTHOR
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Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 19 1999
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