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%I #64 Mar 29 2021 15:09:51
%S 1,0,1,2,4,6,11,16,27,40,63,92,141,202,299,426,614,862,1222,1694,2362,
%T 3242,4456,6054,8229,11072,14891,19872,26477,35050,46320,60866,79827,
%U 104194,135703,176008,227791,293702,377874,484554,620011,790952,1006924
%N Number of unimodal compositions of n+2 where the maximal part appears exactly twice.
%C Old name was: Expansion of a q-series.
%C 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
%C From _Gus Wiseman_, Mar 25 2021: (Start)
%C 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:
%C (1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
%C (2,1) (2,2) (2,3) (2,4) (2,5)
%C (3,1) (3,2) (3,3) (3,4)
%C (1,1,1,1) (4,1) (4,2) (4,3)
%C (1,2,1,1) (5,1) (5,2)
%C (2,1,1,1) (1,2,1,2) (6,1)
%C (1,3,1,1) (1,3,1,2)
%C (2,1,2,1) (1,4,1,1)
%C (2,2,1,1) (2,2,1,2)
%C (3,1,1,1) (2,2,2,1)
%C (1,1,1,1,1,1) (2,3,1,1)
%C (3,1,2,1)
%C (3,2,1,1)
%C (4,1,1,1)
%C (1,2,1,1,1,1)
%C (2,1,1,1,1,1)
%C (End)
%H Alois P. Heinz, <a href="/A114921/b114921.txt">Table of n, a(n) for n = 0..10000</a>
%H S. Fu and D. Tang, <a href="https://arxiv.org/abs/1705.10067">On a generalized crank for k-colored partitions</a>, arXiv:1705.10067 [math.CO], 2017.
%H B. Kim and J. Lovejoy, <a href="https://doi.org/10.1007/s00026-015-0281-x">Ramanujan-type partial theta identities and rank differences for special unimodal sequences</a>, Annals of Combinatorics, 19 (2015), 705-733.
%F 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.
%F 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
%F a(n) = A006330(n) - A001523(n). - _Vaclav Kotesovec_, Jun 22 2015
%F a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (16 * 3^(5/4) * n^(7/4)). - _Vaclav Kotesovec_, Oct 24 2018
%e From _Joerg Arndt_, Jun 10 2013: (Start)
%e There are a(7)=16 such compositions of 7+2=9 where the maximal part appears twice:
%e 01: [ 1 1 1 1 1 2 2 ]
%e 02: [ 1 1 1 1 2 2 1 ]
%e 03: [ 1 1 1 2 2 1 1 ]
%e 04: [ 1 1 1 3 3 ]
%e 05: [ 1 1 2 2 1 1 1 ]
%e 06: [ 1 1 3 3 1 ]
%e 07: [ 1 2 2 1 1 1 1 ]
%e 08: [ 1 2 3 3 ]
%e 09: [ 1 3 3 1 1 ]
%e 10: [ 1 3 3 2 ]
%e 11: [ 1 4 4 ]
%e 12: [ 2 2 1 1 1 1 1 ]
%e 13: [ 2 3 3 1 ]
%e 14: [ 3 3 1 1 1 ]
%e 15: [ 3 3 2 1 ]
%e 16: [ 4 4 1 ]
%e (End)
%t 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. *)
%t wdw[q_]:=And@@Table[q[[i]]>=q[[i+2]],{i,Length[q]-2}];
%t Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],EvenQ[Length[#]]&],wdw]],{n,0,15}] (* _Gus Wiseman_, Mar 25 2021 *)
%o (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))};
%o (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))};
%Y Cf. A226541 (max part appears three times), A188674 (max part m appears m times), A001523 (max part appears any number of times).
%Y Column k=2 of A247255.
%Y A000041 counts weakly increasing (or weakly decreasing) compositions.
%Y A000203 adds up divisors.
%Y A002843 counts compositions with all adjacent parts x <= 2y.
%Y A003242 counts anti-run compositions.
%Y A034008 counts even-length compositions.
%Y A065608 counts even-length compositions with alternating parts equal.
%Y A342528 counts compositions with alternating parts weakly decreasing.
%Y A342532 counts even-length compositions with alternating parts unequal.
%Y Cf. A000726, A001522, A008965, A062968, A064410, A064428, A069916, A070211, A175342, A224958, A342495, A342527.
%K nonn
%O 0,4
%A _Michael Somos_, Jan 07 2006
%E New name from _Joerg Arndt_, Jun 10 2013
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