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Search: id:A000025
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| A000025 |
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Coefficients of the 3rd order mock theta function f(q)
(Formerly M0433 N0164)
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+0
12
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| 1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 82, Examples 4 and 5.
L. A. Dragonette, Some asymptotic formulae for the Mock Theta Series 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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 17, 31.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. (26.11), (26.24).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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a(n) = number of partitions of n with even rank minus number with odd rank. The rank of a partition is its largest part minus the number of parts.
G.f.: 1+Sum_{n>0} (q^(n^2)/Product_{i=1..n}(1+q^i)^2) = (1+4*Sum_{n>0} (-1)^n*q^(n*(3*n+1)/2)/(1+q^n))/Product_{i>0}(1-q^i).
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EXAMPLE
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1 + q - 2*q^2 + 3*q^3 - 3*q^4 + 3*q^5 - 5*q^6 + 7*q^7 - 6*q^8 + 6*q^9 + ...
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MAPLE
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series(1+4*add( (-1)^n*q^(n*(3*n+1)/2)/(1+q^n), n=1..71), q, 71)/series(mul(1-q^i, i=1..71), q, 71);
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MATHEMATICA
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Series[(1+4Sum[(-1)^n q^(n(3n+1)/2)/(1+q^n), {n, 1, 10}])/Sum[(-1)^n q^(n(3n+1)/2), {n, -8, 8}], {q, 0, 100}]
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PROGRAM
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(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(i=1, k, 1 + x^i, 1 + x * O(x^(n - k^2)))^2, 1), n))} /* Michael Somos Sep 02 2007 */
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CROSSREFS
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Other '3rd order' mock theta functions are at A053250, A053251, A053252, A053253, A053254, A053255. See also A000039, A000199.
Sequence in context: A029065 A162157 A060210 this_sequence A036020 A036024 A036029
Adjacent sequences: A000022 A000023 A000024 this_sequence A000026 A000027 A000028
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Entry improved by comments from Dean Hickerson (dean.hickerson(AT)yahoo.com)
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