|
%I #20 Oct 09 2015 17:10:04
%S 1,2,4,6,10,14,20,26,36,46,60,74,94,114,140,166,202,238,284,330,390,
%T 450,524,598,692,786,900,1014,1154,1294,1460,1626,1827,2028,2264,2500,
%U 2780,3060,3384,3708,4088,4468,4904,5340,5844,6348,6920,7492,8148,8804,9544
%N G.f.: 1/((1-x)^2*(1-x^2)*(1-x^4)*(1-x^8)*(1-x^16)).
%C a(n) is the number of partitions of 2*n into powers of 2 less than or equal to 2^5. First differs from A000123 at n=32. - _Alois P. Heinz_, Apr 02 2012
%H Alois P. Heinz, <a href="/A088954/b088954.txt">Table of n, a(n) for n = 0..2000</a>
%H N. J. A. Sloane and J. A. Sellers, <a href="http://arXiv.org/abs/math.CO/0312418">On non-squashing partitions</a>, Discrete Math., 294 (2005), 259-274.
%H <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, -2, 2, -2, 0, 2, 0, -2, 0, 2, -2, 2, 0, -2, 2, -2, 0, 2, -2, 2, 0, -2, 0, 2, 0, -2, 2, -2, 0, 2, -1).
%F a(0)=1, a(1)=2, a(2)=4, a(3)=6, a(4)=10, a(5)=14, a(6)=20, a(7)=26, a(8)=36, a(9)=46, a(10)=60, a(11)=74, a(12)=94, a(13)=114, a(14)=140, a(15)=166, a(16)=202, a(17)=238, a(18)=284, a(19)=330, a(20)=390, a(21)=450, a(22)=524, a(23)=598, a(24)=692, a(25)=786, a(26)=900, a(27)=1014, a(28)=1154, a(29)=1294, a(30)=1460, a(31)=1626, a(n)=2*a(n-1)-2*a(n-3)+ 2*a(n-4)- 2*a(n-5)+ 2*a(n-7)-2*a(n-9)+2*a(n-11)-2*a(n-12)+2*a(n-13)-2*a(n-15)+2*a(n-16)-2*a(n-17)+ 2*a(n-19)- 2*a(n-20)+ 2*a(n-21)-2*a(n-23)+2*a(n-25)-2*a(n-27)+2*a(n-28)-2*a(n-29)+ 2*a(n-31)-a(n-32). - _Harvey P. Dale_, Feb 12 2013
%p f := proc(n,k) option remember; if k > n then RETURN(0); fi; if k= 0 then if n=0 then RETURN(1) else RETURN(0); fi; fi; if k = 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(f(n-1,k)); fi; f(n-1,k)+f(n/2,k-1); end; # present sequence is f(2m,6)
%p GFF := k->x^(2^(k-2))/((1-x)*mul((1-x^(2^j)),j=0..k-2)); # present g.f. is GFF(6)/x^16
%p a:= proc(n) local m, r; m:= iquo(n, 16, 'r'); r:= r+1; [1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, 94, 114, 140, 166][r] +(((((128/5*m +8*(15+r))*m +(228 +[0, 32, 68, 104, 144, 184, 228, 272, 320, 368, 420, 472, 528, 584, 644, 704][r]))*m +(172 +[0, 43, 98, 153, 223, 293, 378, 463, 566, 669, 790, 911, 1053, 1195, 1358, 1521][r]))*m +(247/5 +[0, 22, 55, 88, 138, 188, 255, 322, 415, 508, 627, 746, 900, 1054, 1243, 1432][r]))*m)/3 end: seq(a(n), n=0..60); # _Alois P. Heinz_, Apr 17 2009
%t CoefficientList[Series[1/((1-x)^2(1-x^2)(1-x^4)(1-x^8)(1-x^16)),{x,0,70}],x] (* or *) LinearRecurrence[{2,0,-2,2,-2,0,2,0,-2,0,2,-2,2,0,-2,2,-2,0,2,-2,2,0,-2,0,2,0,-2,2,-2,0,2,-1},{1,2,4,6,10,14,20,26,36,46,60,74,94,114,140,166,202,238,284,330,390,450,524,598,692,786,900,1014,1154,1294,1460,1626},70](* _Harvey P. Dale_, Feb 12 2013 *)
%Y See A000027, A002620, A008804, A088932, A000123 for similar sequences.
%Y Column k=5 of A181322.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Dec 02 2003
|
