A053251 Coefficients of the '3rd-order' mock theta function psi(q).
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
Offset: 0
Examples
q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ... From _Seiichi Manyama_, Mar 17 2018: (Start) n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m) --+--------------------------+------------------------- 1 | (1) | (1) 2 | (2) | (2) 3 | (3) | (3) 4 | (4) | (4) | (1, 3) | (1, 3/2) 5 | (5) | (5) | (1, 4) | (1, 2) 6 | (6) | (6) | (1, 5) | (1, 5/2) 7 | (7) | (7) | (1, 6) | (1, 3) | (2, 5) | (2, 5/2) 8 | (8) | (8) | (1, 7) | (1, 7/2) | (2, 6) | (2, 3) 9 | (9) | (9) | (1, 8) | (1, 4) | (2, 7) | (2, 7/2) | (1, 3, 5) | (1, 3/2, 5/3) (End)
References
- 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- Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
- Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
- George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.
Crossrefs
Programs
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Maple
f:=n->q^(n^2)/mul((1-q^(2*i+1)),i=0..n-1); add(f(i),i=1..6); # second Maple program: b:= proc(n, i) option remember; (s-> `if`(n>s, 0, `if`(n=s, 1, b(n, i-1)+b(n-i, min(n-i, i-1)))))(i*(i+1)/2) end: a:= n-> `if`(n=0, 0, add(b(j, min(j, n-2*j-1)), j=0..iquo(n, 2))): seq(a(n), n=0..80); # Alois P. Heinz, May 17 2018 -
Mathematica
Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 100}] (* Second program: *) b[n_, i_] := b[n, i] = Function[s, If[n > s, 0, If[n == s, 1, b[n, i - 1] + b[n - i, Min[n - i, i - 1]]]]][i*(i + 1)/2]; a[n_] := If[n==0, 0, Sum[b[j, Min[j, n-2*j-1]], {j, 0, Quotient[n, 2]}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 17 2018, after Alois P. Heinz *) -
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 */
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*Product_{j=1..k-1} (1+q^(2*j)) (see the Fine reference, p. 58, Eq. (26,53)). - Emeric Deutsch, Mar 08 2006
a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 09 2019
Extensions
More terms from Emeric Deutsch, Mar 08 2006
Comments