

A053260


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


13



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, 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, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 29, 30, 32, 32, 34, 36, 36, 39, 40, 41, 44, 45, 46
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OFFSET

0,14


COMMENTS

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.
Strongly unimodal compositions with first part 1 and each upstep is by at most 1 (leftsmoothness); 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]


REFERENCES

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
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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22
George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274304


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..10000


FORMULA

G.f.: psi_0(q) = sum(n>=0, q^((n+1)*(n+2)/2) * (1+q)*(1+q^2)*...*(1+q^n) ).


EXAMPLE

From Joerg Arndt, Dec 09 2012: (Start)
The a(42)=8 strongly unimodal leftsmooth compositions are
[ #] composition
[ 1] [ 1 2 3 4 5 6 7 5 4 3 2 ]
[ 2] [ 1 2 3 4 5 6 7 6 4 3 1 ]
[ 3] [ 1 2 3 4 5 6 7 6 5 2 1 ]
[ 4] [ 1 2 3 4 5 6 7 6 5 3 ]
[ 5] [ 1 2 3 4 5 6 7 8 3 2 1 ]
[ 6] [ 1 2 3 4 5 6 7 8 4 2 ]
[ 7] [ 1 2 3 4 5 6 7 8 5 1 ]
[ 8] [ 1 2 3 4 5 6 7 8 6 ]
(End)


MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i1)+`if`(i>n, 0, b(ni, i1))))
end:
a:= proc(n) local h, k, m, r;
m, r:= floor((sqrt(n*8+1)1)/2), 0;
for k from m by 1 do h:= k*(k+1);
if h<=n then break fi;
r:= r+b(nh/2, k1)
od: r
end:
seq(a(n), n=0..100); # Alois P. Heinz, Aug 02 2013


MATHEMATICA

Series[Sum[q^((n+1)(n+2)/2) Product[1+q^k, {k, 1, n}], {n, 0, 12}], {q, 0, 100}]


PROG

(PARI)
N = 66; x = 'x + O('x^N);
gf = sum(n=1, N, x^(n*(n+1)/2) * prod(k=1, n1, 1+x^k) ) + 'c0;
v = Vec(gf); v[1]='c0; v
/* Joerg Arndt, Apr 21 2013 */


CROSSREFS

Other '5th order' mock theta functions are at A053256, A053257, A053258, A053259, A053261, A053262, A053263, A053264, A053265, A053266, A053267.
Sequence in context: A102382 A024890 A007895 * A140223 A014643 A236265
Adjacent sequences: A053257 A053258 A053259 * A053261 A053262 A053263


KEYWORD

nonn,easy


AUTHOR

Dean Hickerson, Dec 19 1999


STATUS

approved



