A053260 Coefficients of the '5th-order' mock theta function psi_0(q).

0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 3, 4, 4, 4, 5, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9, 8, 9, 10, 9, 11, 11, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 29, 30, 32, 32, 34, 36, 36, 39, 40, 41, 44, 45, 46

Offset: 0

Author

Dean Hickerson, Dec 19 1999

Keywords

• nonn
• easy

Comments

Number of partitions of n such that each part occurs at most twice, the largest part is unique and if k occurs as a part then all smaller positive integers occur.

Strongly unimodal compositions with first part 1 and each up-step is by at most 1 (left-smoothness); with this interpretation one should set a(0)=1; see example. Replacing "strongly" by "weakly" in the condition gives A001524. Dropping the requirement of unimodality gives A005169. [Joerg Arndt, Dec 09 2012]

Examples

			From _Joerg Arndt_, Dec 09 2012: (Start)
The a(42)=8 strongly unimodal left-smooth compositions are
[ #]       composition
[ 1]    [ 1 2 3 4 5 6 7 5 4 3 2 ]
[ 2]    [ 1 2 3 4 5 6 7 6 4 3 1 ]
[ 3]    [ 1 2 3 4 5 6 7 6 5 2 1 ]
[ 4]    [ 1 2 3 4 5 6 7 6 5 3 ]
[ 5]    [ 1 2 3 4 5 6 7 8 3 2 1 ]
[ 6]    [ 1 2 3 4 5 6 7 8 4 2 ]
[ 7]    [ 1 2 3 4 5 6 7 8 5 1 ]
[ 8]    [ 1 2 3 4 5 6 7 8 6 ]
(End)
		

References

• 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• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113-134.
• George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.
• George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304.

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053261, A053262, A053263, A053264, A053265, A053266, A053267.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1)+`if`(i>n, 0, b(n-i, i-1))))
    end:
a:= proc(n) local h, k, m, r;
      m, r:= floor((sqrt(n*8+1)-1)/2), 0;
      for k from m by -1 do h:= k*(k+1);
        if h<=n then break fi;
        r:= r+b(n-h/2, k-1)
      od: r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 02 2013

Mathematica

Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}], {n, 0, 12}], {q, 0, 100}]
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-1] ] ]]; a[n_] := Module[{h, k, m, r}, {m, r} = {Floor[(Sqrt[n*8+1]-1)/2], 0}; For[k = m, True, k--, h = k*(k+1); If[h <= n, Break[]]; r = r + b[n-h/2, k-1]]; r]; Table[ a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 09 2015, after Alois P. Heinz *)

PARI

N = 66;  x = 'x + O('x^N);
gf = sum(n=1,N, x^(n*(n+1)/2) * prod(k=1,n-1,1+x^k) ) + 'c0;
v = Vec(gf); v[1]-='c0; v
/* Joerg Arndt, Apr 21 2013 */

Formula

G.f.: psi_0(q) = Sum_{n>=0} q^((n+1)*(n+2)/2) * (1+q)*(1+q^2)*...*(1+q^n).

a(n) ~ exp(Pi*sqrt(n/15)) / (2*5^(1/4)*sqrt(phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019