Search: id:A000025
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%I A000025 M0433 N0164
%S A000025 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,
%T A000025 68,78,82,89,104,118,123,131,154,171,179,197,221,245,262,279,314,349,
%U A000025 369,398,446,486,515,557,614,671,715,767,845,920,977,1046,1148,1244
%V A000025 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,
%W A000025 -68,78,-82,89,-104,118,-123,131,-154,171,-179,197,-221,245,-262,279,-314,
               349,
%X A000025 -369,398,-446,486,-515,557,-614,671,-715,767,-845,920,-977,1046,-1148,
               1244
%N A000025 Coefficients of the 3rd order mock theta function f(q)
%D A000025 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000025 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000025 G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 82, 
               Examples 4 and 5.
%D A000025 L. A. Dragonette, Some asymptotic formulae for the Mock Theta Series 
               of Ramanujan, Trans. Amer. Math. Soc., 72 (1952), 474-500.
%D A000025 Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355
%D A000025 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, 
               Narosa Publishing House, New Delhi, 1988, pp. 17, 31.
%D A000025 George N. Watson, The final problem: an account of the mock theta functions, 
               J. London Math. Soc., 11 (1936) 55-80
%D A000025 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 55, Eq. (26.11), (26.24).
%H A000025 T. D. Noe, <a href="b000025.txt">Table of n, a(n) for n = 0..1000</a>
%H A000025 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MockThetaFunction.html">Link to a section of The World of Mathematics.</
               a>
%F A000025 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.
%F A000025 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).
%e A000025 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 
               + ...
%p A000025 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);
%t A000025 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}]
%o A000025 (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 */
%Y A000025 Other '3rd order' mock theta functions are at A053250, A053251, A053252, 
               A053253, A053254, A053255. See also A000039, A000199.
%Y A000025 Sequence in context: A029065 A162157 A060210 this_sequence A036020 A036024 
               A036029
%Y A000025 Adjacent sequences: A000022 A000023 A000024 this_sequence A000026 A000027 
               A000028
%K A000025 sign,easy,nice
%O A000025 0,3
%A A000025 N. J. A. Sloane (njas(AT)research.att.com).
%E A000025 Entry improved by comments from Dean Hickerson (dean.hickerson(AT)yahoo.com)

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