The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A122129 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A122129

Expansion of 1 + Sum_{k>0} x^k^2/((1-x)(1-x^2)...(1-x^(2k))).
(history; published version)

#55 by N. J. A. Sloane at Fri Mar 11 12:41:48 EST 2022

STATUS

proposed

approved

#54 by Michel Marcus at Thu Feb 24 01:16:59 EST 2022

STATUS

editing

proposed

#53 by Michel Marcus at Thu Feb 24 01:16:52 EST 2022

REFERENCES

Watson, G. N. (1937), "The Mock Theta Functions (2)", Proceedings of the London Mathematical Society, s2-42: 274-304, doi:10.1112/plms/s2-42.1.274

LINKS

George N. Watson, <a href="https://doi.org/10.1112/plms/s2-42.1.274">The mock theta functions (2)</a>, Proc. London Math. Soc., series 2, 42 (1937) 274-304.

STATUS

proposed

editing

#52 by Jon E. Schoenfield at Thu Feb 24 00:18:28 EST 2022

STATUS

editing

proposed

#51 by Jon E. Schoenfield at Thu Feb 24 00:18:21 EST 2022

COMMENTS

a(n) = number of SE partitions of n, for n >= 1; see A237981. _- _Clark Kimberling_, Mar 19 2014

FORMULA

Let f(n) = 1/Product_{k >= 0} (1 - q^(20k+n)). Then g.f. is f(1)*f(3)*f(4)*f(5)*f(7)*f(9)*f(11)*f(13)*f(15)*f(16)*f(17)*f(19). - N. J. A. Sloane, Mar 19 2012.

a(n) = is the number of partitions of n into parts that are either odd or == +/-4 (mod 20). - Michael Somos, Jun 28 2015

STATUS

proposed

editing

#50 by Gus Wiseman at Wed Feb 23 20:34:47 EST 2022

STATUS

editing

proposed

#49 by Gus Wiseman at Wed Feb 23 20:27:56 EST 2022

COMMENTS

This appears to be the number of integer partitions of n with every other pair of adjacent parts strictly decreasing, as in the pattern a > b >= c > d >= e for a partition (a, b, c, d, e). For example, the a(1) = 1 through a(9) = 12 partitions are:

STATUS

proposed

editing

Discussion

Wed Feb 23

20:34

Gus Wiseman: hopefully a little clearer

#48 by Gus Wiseman at Tue Feb 22 05:51:55 EST 2022

STATUS

editing

proposed

#47 by Gus Wiseman at Tue Feb 22 05:51:25 EST 2022

COMMENTS

The even-length case is A351008. The odd-length case appears to be A122130. If the opposite Swapping strictly and weakly decreasing relations are strict we seem appears to get give A122135. The alternately unequal and equal case is A351006, strict A035457, opposite A351005, even-length A351007.

#46 by Gus Wiseman at Tue Feb 22 05:32:18 EST 2022

CROSSREFS

Cf. A000009, A000041, A000070, A096441, A122134, A351004.