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%I #70 Aug 08 2022 09:39:41
%S 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,
%T 42,46,51,57,62,68,76,83,91,101,109,120,132,143,156,171,186,202,221,
%U 239,259,283,306,331,360,388,420,455,490,529,572,616,663,716,769,827
%N Coefficients of the '3rd-order' mock theta function psi(q)
%C 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.
%C Number of ways to express n as a partial sum 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+3+1+1. - _Jon Perry_, Jan 01 2004
%C Also number of partitions of n such that the 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
%C Number of partitions (d1,d2,...,dm) of n such that 0 < d1/1 < d2/2 < ... < dm/m. - _Seiichi Manyama_, Mar 17 2018
%C For _Emeric Deutsch_'s comment above, (1) this appears to be an alternately equal case of A122130, (2) the ordered version (compositions) is A239327, (3) allowing any length gives A351006, (4) the even-length version is A351007. - _Gus Wiseman_, Feb 25 2022
%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 55, Eq. (26.13).
%D Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 31.
%H Alois P. Heinz, <a href="/A053251/b053251.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H Leila A. Dragonette, <a href="http://dx.doi.org/10.1090/S0002-9947-1952-0049927-8">Some asymptotic formulas for the mock theta series of Ramanujan</a>, Trans. Amer. Math. Soc., 72 (1952) 474-500.
%H George N. Watson, <a href="https://doi.org/10.1112/jlms/s1-11.1.55">The final problem: an account of the mock theta functions</a>, J. London Math. Soc., 11 (1936) 55-80.
%F G.f.: psi(q) = Sum_{n>=1} q^(n^2) / ( (1-q)*(1-q^3)*...*(1-q^(2*n-1)) ).
%F 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
%F a(n) ~ exp(Pi*sqrt(n/6)) / (4*sqrt(n)). - _Vaclav Kotesovec_, Jun 09 2019
%e q + q^2 + q^3 + 2*q^4 + 2*q^5 + 2*q^6 + 3*q^7 + 3*q^8 + 4*q^9 + ...
%e From _Seiichi Manyama_, Mar 17 2018: (Start)
%e n | Partition (d1,d2,...,dm) | (d1/1, d2/2, ... , dm/m)
%e --+--------------------------+-------------------------
%e 1 | (1) | (1)
%e 2 | (2) | (2)
%e 3 | (3) | (3)
%e 4 | (4) | (4)
%e | (1, 3) | (1, 3/2)
%e 5 | (5) | (5)
%e | (1, 4) | (1, 2)
%e 6 | (6) | (6)
%e | (1, 5) | (1, 5/2)
%e 7 | (7) | (7)
%e | (1, 6) | (1, 3)
%e | (2, 5) | (2, 5/2)
%e 8 | (8) | (8)
%e | (1, 7) | (1, 7/2)
%e | (2, 6) | (2, 3)
%e 9 | (9) | (9)
%e | (1, 8) | (1, 4)
%e | (2, 7) | (2, 7/2)
%e | (1, 3, 5) | (1, 3/2, 5/3) (End)
%p f:=n->q^(n^2)/mul((1-q^(2*i+1)),i=0..n-1); add(f(i),i=1..6);
%p # second Maple program:
%p b:= proc(n, i) option remember; (s-> `if`(n>s, 0, `if`(n=s, 1,
%p b(n, i-1)+b(n-i, min(n-i, i-1)))))(i*(i+1)/2)
%p end:
%p a:= n-> `if`(n=0, 0, add(b(j, min(j, n-2*j-1)), j=0..iquo(n, 2))):
%p seq(a(n), n=0..80); # _Alois P. Heinz_, May 17 2018
%t Series[Sum[q^n^2/Product[1-q^(2k-1), {k, 1, n}], {n, 1, 10}], {q, 0, 100}]
%t (* Second program: *)
%t 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];
%t a[n_] := If[n==0, 0, Sum[b[j, Min[j, n-2*j-1]], {j, 0, Quotient[n, 2]}]];
%t Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Jun 17 2018, after _Alois P. Heinz_ *)
%o (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_
%o (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 */
%Y Other '3rd-order' mock theta functions are at A000025, A053250, A053252, A053253, A053254, A053255.
%Y Cf. A003475.
%Y Cf. A035363, A035457, A122129, A122130, A122134, A122135, A351003, A351005.
%K nonn,easy
%O 0,5
%A _Dean Hickerson_, Dec 19 1999
%E More terms from _Emeric Deutsch_, Mar 08 2006
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