OFFSET
0,4
COMMENTS
Old name was: Expansion of a q-series.
a(n) is also the number of 2-colored partitions of n with the same number of parts in each color. - Shishuo Fu, May 30 2017
From Gus Wiseman, Mar 25 2021: (Start)
Also the number of even-length compositions of n with alternating parts weakly decreasing. Allowing odd lengths also gives A342528. The version with alternating parts strictly decreasing appears to be A064428. The a(2) = 1 through a(7) = 16 compositions are:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5)
(3,1) (3,2) (3,3) (3,4)
(1,1,1,1) (4,1) (4,2) (4,3)
(1,2,1,1) (5,1) (5,2)
(2,1,1,1) (1,2,1,2) (6,1)
(1,3,1,1) (1,3,1,2)
(2,1,2,1) (1,4,1,1)
(2,2,1,1) (2,2,1,2)
(3,1,1,1) (2,2,2,1)
(1,1,1,1,1,1) (2,3,1,1)
(3,1,2,1)
(3,2,1,1)
(4,1,1,1)
(1,2,1,1,1,1)
(2,1,1,1,1,1)
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000
S. Fu and D. Tang, On a generalized crank for k-colored partitions, arXiv:1705.10067 [math.CO], 2017.
B. Kim and J. Lovejoy, Ramanujan-type partial theta identities and rank differences for special unimodal sequences, Annals of Combinatorics, 19 (2015), 705-733.
FORMULA
G.f.: 1 + Sum_{k>0} (x^k / ((1-x)(1-x^2)...(1-x^k)))^2 = (1 + Sum_{k>0} 2 (-1)^k x^((k^2+k)/2) ) / (Product_{k>0} (1 - x^k))^2.
G.f.: 1 + x*(1 - G(0))/(1-x) where G(k) = 1 - x/(1-x^(k+1))^2/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (16 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Oct 24 2018
EXAMPLE
From Joerg Arndt, Jun 10 2013: (Start)
There are a(7)=16 such compositions of 7+2=9 where the maximal part appears twice:
01: [ 1 1 1 1 1 2 2 ]
02: [ 1 1 1 1 2 2 1 ]
03: [ 1 1 1 2 2 1 1 ]
04: [ 1 1 1 3 3 ]
05: [ 1 1 2 2 1 1 1 ]
06: [ 1 1 3 3 1 ]
07: [ 1 2 2 1 1 1 1 ]
08: [ 1 2 3 3 ]
09: [ 1 3 3 1 1 ]
10: [ 1 3 3 2 ]
11: [ 1 4 4 ]
12: [ 2 2 1 1 1 1 1 ]
13: [ 2 3 3 1 ]
14: [ 3 3 1 1 1 ]
15: [ 3 3 2 1 ]
16: [ 4 4 1 ]
(End)
MATHEMATICA
max = 50; s = (1+Sum[2*(-1)^k*q^(k(k+1)/2), {k, 1, max}])/QPochhammer[q]^2+ O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, from 1st g.f. *)
wdw[q_]:=And@@Table[q[[i]]>=q[[i+2]], {i, Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], EvenQ[Length[#]]&], wdw]], {n, 0, 15}] (* Gus Wiseman, Mar 25 2021 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, n\2, x^(2*k) / prod(i=1, k, 1 - x^i, 1 + x * O(x^n))^2), n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum(k=1, sqrtint(8*n + 1)\2, 2*(-1)^k * x^((k^2+k)/2), 1 + A) / eta(x + A)^2, n))};
CROSSREFS
Cf. A226541 (max part appears three times), A188674 (max part m appears m times), A001523 (max part appears any number of times).
Column k=2 of A247255.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A000203 adds up divisors.
A002843 counts compositions with all adjacent parts x <= 2y.
A003242 counts anti-run compositions.
A034008 counts even-length compositions.
A065608 counts even-length compositions with alternating parts equal.
A342528 counts compositions with alternating parts weakly decreasing.
A342532 counts even-length compositions with alternating parts unequal.
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 07 2006
EXTENSIONS
New name from Joerg Arndt, Jun 10 2013
STATUS
approved