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Coefficients of the '5th-order' mock theta function psi_1(q).
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%I #38 Feb 02 2021 21:57:33

%S 1,1,1,1,1,1,2,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,6,7,7,7,8,

%T 8,9,10,10,10,11,11,12,13,13,14,15,16,16,17,18,19,20,20,21,23,24,25,

%U 26,27,28,30,31,32,34,35,37,39,40,41,44,45,47,50,51,53,56,58,60,63,65

%N Coefficients of the '5th-order' mock theta function psi_1(q).

%C Number of partitions of n such that each part occurs at most twice and if k occurs as a part then all smaller positive integers occur.

%C Strictly unimodal compositions with rising range 1, 2, 3, ..., m where m is the largest part and distinct parts in the falling range (this follows trivially from the comment above). [_Joerg Arndt_, Mar 26 2014]

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%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.

%H Vaclav Kotesovec, <a href="/A053261/b053261.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%H 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 George E. Andrews and Frank G. Garvan, <a href="http://dx.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 William J. Keith, <a href="https://arxiv.org/abs/1911.04755">Partitions into parts simultaneously regular, distinct, and/or flat</a>, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence.

%H George N. Watson, <a href="http://plms.oxfordjournals.org/content/s2-42/1/274.extract">The mock theta functions (2)</a>, Proc. London Math. Soc., series 2, 42 (1937) 274-304.

%F G.f.: psi_1(q) = Sum_{n>=0} q^(n*(n+1)/2) * Product_{k=1..n} (1 + q^k).

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

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p `if`(i>n, 0, add(b(n-i*j, i+1), j=1..min(2, n/i))))

%p end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=0..100); # _Alois P. Heinz_, Mar 26 2014

%t Series[Sum[q^(n(n+1)/2) Product[1+q^k, {k, 1, n}], {n, 0, 13}], {q, 0, 100}]

%t (* Second program: *)

%t b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, Sum[b[n - i*j, i + 1], {j, 1, Min[2, n/i]}]]];

%t a[n_] := b[n, 1];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Jun 09 2018, after _Alois P. Heinz_ *)

%t nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1)/2) * Product[1+x^j, {j, 1, k}], {k, 0, Floor[Sqrt[2*nmax]]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 12 2019 *)

%o (PARI) N = 66; x = 'x + O('x^N); gf = sum(n=0,N, x^(n*(n+1)/2) * prod(k=1,n,1+x^k) ); v = Vec(gf) /* _Joerg Arndt_, Apr 21 2013 */

%Y Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053262, A053263, A053264, A053265, A053266, A053267.

%K nonn,easy

%O 0,7

%A _Dean Hickerson_, Dec 19 1999