OFFSET
0,14
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]
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.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.
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
EXAMPLE
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)
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
STATUS
approved