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A053251
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Coefficients of the '3rd order' mock theta function psi(q)
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11
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0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 19, 22, 24, 27, 31, 34, 37, 42, 46, 51, 57, 62, 68, 76, 83, 91, 101, 109, 120, 132, 143, 156, 171, 186, 202, 221, 239, 259, 283, 306, 331, 360, 388, 420, 455, 490, 529, 572, 616, 663, 716, 769, 827
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OFFSET
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0,5
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COMMENTS
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Number of different partial sums of 1+[1,3]+[1,5]+[1,7]+[1,9]+... E.g. a(6)=2 because we have 6=1+1+1+1+1+1=1+1+3+1 - Jon Perry, Jan 01 2004
Also number of partitions of n such that largest part occurs exactly once and all the other parts occur exactly twice. Example: a(9)=4 because we have [9],[7,1,1],[5,2,2] and [3,2,2,1,1]. - Emeric Deutsch, Mar 08 2006
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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
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.13).
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
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FORMULA
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G.f.: psi(q) = sum for n >= 1 of q^n^2/((1-q)(1-q^3)...(1-q^(2n-1)))
a(n) = number of partitions of n into odd parts such that if a number occurs as a part then so do all smaller positive odd numbers
G.f.=sum(q^k*product(1+q^(2j), j=1..k-1), k=1..infinity), (see the Fine reference, p. 58, Eq. (26,53)). - Emeric Deutsch, Mar 08 2006
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EXAMPLE
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q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ...
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MAPLE
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f:=n->q^(n^2)/mul((1-q^(2*i+1)), i=0..n-1); add(f(i), i=1..6);
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MATHEMATICA
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Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 100}]
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PROG
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(PARI) { n=20; v=vector(n); for (i=1, n, v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]+2*i-1)); c=vector(n); for (i=1, n, for (j=1, 2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } (Jon Perry)
(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-1)) + O(x^(n-(k-1)^2+1))), n))} /* Michael Somos Sep 04 2007 */
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CROSSREFS
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Other '3rd order' mock theta functions are at A000025, A053250, A053252, A053253, A053254, A053255.
Cf. A003475.
Sequence in context: A029148 A067842 A164066 * A090184 A174575 A029057
Adjacent sequences: A053248 A053249 A053250 * A053252 A053253 A053254
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KEYWORD
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nonn,easy
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AUTHOR
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Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 19 1999
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EXTENSIONS
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More terms from Emeric Deutsch, Mar 08 2006
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STATUS
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approved
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