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A053251 Coefficients of the '3rd order' mock theta function psi(q) 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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.

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

REFERENCES

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31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

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

FORMULA

G.f.: psi(q) = sum(n>=1, q^(n^2) / ( (1-q)*(1-q^3)*...*(1-q^(2*n-1)) ) ).

G.f.: sum(k>=1, q^k*prod(j=1..k-1, 1+q^(2*j) ) ), (see the Fine reference, p. 58, Eq. (26,53)). - Emeric Deutsch, Mar 08 2006

EXAMPLE

q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ...

MAPLE

f:=n->q^(n^2)/mul((1-q^(2*i+1)), i=0..n-1); add(f(i), i=1..6);

MATHEMATICA

Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 100}]

PROG

(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 */

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Dean Hickerson, Dec 19 1999

EXTENSIONS

More terms from Emeric Deutsch, Mar 08 2006

STATUS

approved

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Last modified September 19 01:50 EDT 2014. Contains 246967 sequences.