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

%I A053260 #39 Jan 08 2024 14:28:42
%S A053260 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,
%T A053260 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,
%U A053260 19,19,21,22,22,24,25,25,27,28,29,30,32,32,34,36,36,39,40,41,44,45,46
%N A053260 Coefficients of the '5th-order' mock theta function psi_0(q).
%C A053260 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.
%C A053260 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]
%D A053260 Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354-355.
%D A053260 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.
%H A053260 Alois P. Heinz, <a href="/A053260/b053260.txt">Table of n, a(n) for n = 0..10000</a>
%H A053260 George E. Andrews, <a href="http://dx.doi.org/10.1090/S0002-9947-1986-0814916-2">The fifth and seventh order mock theta functions</a>, Trans. Amer. Math. Soc., 293 (1986) 113-134.
%H A053260 George E. Andrews and Frank G. Garvan, <a href="https://doi.org/10.1016/0001-8708(89)90070-4">Ramanujan's "lost" notebook VI: The mock theta conjectures</a>, Advances in Mathematics, 73 (1989) 242-255.
%H A053260 George N. Watson, <a href="https://doi.org/10.1112/plms/s2-42.1.274">The mock theta functions (2)</a>, Proc. London Math. Soc., series 2, 42 (1937) 274-304.
%F A053260 G.f.: psi_0(q) = Sum_{n>=0} q^((n+1)*(n+2)/2) * (1+q)*(1+q^2)*...*(1+q^n).
%F A053260 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
%e A053260 From _Joerg Arndt_, Dec 09 2012: (Start)
%e A053260 The a(42)=8 strongly unimodal left-smooth compositions are
%e A053260 [ #]       composition
%e A053260 [ 1]    [ 1 2 3 4 5 6 7 5 4 3 2 ]
%e A053260 [ 2]    [ 1 2 3 4 5 6 7 6 4 3 1 ]
%e A053260 [ 3]    [ 1 2 3 4 5 6 7 6 5 2 1 ]
%e A053260 [ 4]    [ 1 2 3 4 5 6 7 6 5 3 ]
%e A053260 [ 5]    [ 1 2 3 4 5 6 7 8 3 2 1 ]
%e A053260 [ 6]    [ 1 2 3 4 5 6 7 8 4 2 ]
%e A053260 [ 7]    [ 1 2 3 4 5 6 7 8 5 1 ]
%e A053260 [ 8]    [ 1 2 3 4 5 6 7 8 6 ]
%e A053260 (End)
%p A053260 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A053260        b(n, i-1)+`if`(i>n, 0, b(n-i, i-1))))
%p A053260     end:
%p A053260 a:= proc(n) local h, k, m, r;
%p A053260       m, r:= floor((sqrt(n*8+1)-1)/2), 0;
%p A053260       for k from m by -1 do h:= k*(k+1);
%p A053260         if h<=n then break fi;
%p A053260         r:= r+b(n-h/2, k-1)
%p A053260       od: r
%p A053260     end:
%p A053260 seq(a(n), n=0..100);  # _Alois P. Heinz_, Aug 02 2013
%t A053260 Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}], {n, 0, 12}], {q, 0, 100}]
%t A053260 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_ *)
%o A053260 (PARI)
%o A053260 N = 66;  x = 'x + O('x^N);
%o A053260 gf = sum(n=1,N, x^(n*(n+1)/2) * prod(k=1,n-1,1+x^k) ) + 'c0;
%o A053260 v = Vec(gf); v[1]-='c0; v
%o A053260 /* _Joerg Arndt_, Apr 21 2013 */
%Y A053260 Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053261, A053262, A053263, A053264, A053265, A053266, A053267.
%K A053260 nonn,easy
%O A053260 0,14
%A A053260 _Dean Hickerson_, Dec 19 1999