

A053261


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


12



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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22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

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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
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) * prod(k=1..n, 1 + q^k ) ).


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}]


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 '5th order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053262, A053263, A053264, A053265, A053266, A053267.
Sequence in context: A165640 A082892 A025839 * A123584 A112689 A190353
Adjacent sequences: A053258 A053259 A053260 * A053262 A053263 A053264


KEYWORD

nonn,easy


AUTHOR

Dean Hickerson, Dec 19 1999


STATUS

approved



