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
Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 < d2/2 < ... < dm/m. - Seiichi Manyama, Mar 17 2018
From Gus Wiseman, Feb 22 2022: (Start)
Also the number of odd-length integer partitions of n into parts that are alternately unequal and equal. For example, the a(1) = 1 through a(9) = 7 partitions are (A..D = 10..13):
1 2 3 4 5 6 7 8 9 A B C D
211 311 411 322 422 522 433 533 633 544
511 611 711 622 722 822 733
32211 811 911 A11 922
42211 52211 43311 B11
62211 53311
72211
This appears to be the alternately equal case of A122130.
The ordered version (compositions) is A239327.
Allowing any length gives A351006.
The even-length version is A351007.
(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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31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.
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
a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 09 2019
EXAMPLE
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)
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 *)
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&And@@Table[If[EvenQ[i], #[[i]]==#[[i+1]], #[[i]]!=#[[i+1]]], {i, Length[#]-1}]&]], {n, 0, 30}] (* Gus Wiseman, Feb 22 2022 *)
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
Cf. A003475.
KEYWORD
nonn,easy
AUTHOR
Dean Hickerson, Dec 19 1999
EXTENSIONS
More terms from Emeric Deutsch, Mar 08 2006
STATUS
proposed