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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 (list; graph; refs; listen; history; text; internal format)
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. 354-355

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 19, 21, 22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) 113-134

George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255

George N. Watson, The mock theta functions (2), Proc. London Math. Soc., series 2, 42 (1937) 274-304

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(n-i*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

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Last modified September 19 21:44 EDT 2014. Contains 246979 sequences.