

A053261


Coefficients of the '5thorder' mock theta function psi_1(q).


17



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, 8, 9, 10, 10, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 34, 35, 37, 39, 40, 41, 44, 45, 47, 50, 51, 53, 56, 58, 60, 63, 65
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OFFSET

0,7


COMMENTS

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.
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]


REFERENCES

Srinivasa Ramanujan, Collected Papers, Chelsea, New York, 1962, pp. 354355.
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22.


LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
George E. Andrews, The fifth and seventh order mock theta functions, Trans. Amer. Math. Soc., 293 (1986) 113134.
George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242255.
William J. Keith, Partitions into parts simultaneously regular, distinct, and/or flat, Proceedings of CANT 2016; arXiv:1911.04755 [math.CO], 2019. Mentions this sequence.
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274304.


FORMULA

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


MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i>n, 0, add(b(ni*j, i+1), j=1..min(2, n/i))))
end:
a:= n> b(n, 1):
seq(a(n), n=0..100); # Alois P. Heinz, Mar 26 2014


MATHEMATICA

Series[Sum[q^(n(n+1)/2) Product[1+q^k, {k, 1, n}], {n, 0, 13}], {q, 0, 100}]
(* Second program: *)
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]}]]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 100}] (* JeanFrançois Alcover, Jun 09 2018, after Alois P. Heinz *)
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 *)


PROG

(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 */


CROSSREFS

Other '5thorder' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053262, A053263, A053264, A053265, A053266, A053267.
Sequence in context: A165640 A082892 A025839 * A123584 A291983 A112689
Adjacent sequences: A053258 A053259 A053260 * A053262 A053263 A053264


KEYWORD

nonn,easy


AUTHOR

Dean Hickerson, Dec 19 1999


STATUS

approved



