1,7

Useful in the creation of plane partitions with C3 or C3v symmetry

The function T[m,a,b] used here gives the partitions of n whose Ferrers plot fits within an a X b box.

T. D. Noe, Table of n, a(n) for n=1..1000

a(n) = p(n-1)-p(n-2)-p(n-5)+p(n-7)+... +(-1)^k*(p(n-(3*k^2-k)/2)-p(n-(3*k^2+k)/2))+..., where p() is A000041(). E.g. A047993 a(20) = p(19)-p(18)-p(15)+p(13)+p(8)-p(5) = 490-385-176+101+22-7 = 45. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 04 2004

G.f.: Sum_{k>0}((-1)^k*(x^((3*k^2+k)/2)-x^((3*k^2-k)/2)))/Product_{k>0}(1-x^k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 05 2004

{5,4,1,1,1} is a balanced partition of 12 because the first element is 5, and the length is 5.

with(combinat): for n from 1 to 36 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]=nops(P[j]) then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n], n=1..36); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 11 2004

Table[ Count[Partitions[n], par_List/; First[par]===Length[par]], {n, 12}] or recur: Sum[T[n-(2m-1), m-1, m-1], {m, Ceiling[Sqrt[n]], Floor[(n+1)/2]}] with T[m_, a_, b_]/; b < a := T[m, b, a]; T[m_, a_, b_]/; m > a*b := 0; T[m_, a_, b_]/; (2m > a*b) := T[a*b-m, a, b]; T[m_, 1, b_] := If[b < m, 0, 1]; T[0, _, _] := 1; T[m_, a_, b_] := T[m, a, b]=Sum[T[m-a*i, a-1, b-i], {i, 0, Floor[m/a]}];

Table[Sum[ -(-1)^k*(p[n-(3*k^2-k)/2] - p[n-(3*k^2+k)/2]), {k, 1, Floor[(1+Sqrt[1+24*n])/6]}] /. p -> PartitionsP, {n, 1, 64}] (from Wouter Meeussen)

Cf. A063995, A064173, A064174.

Sequence in context: A069745 A112199 A059851 this_sequence A033177 A095401 A073369

Adjacent sequences: A047990 A047991 A047992 this_sequence A047994 A047995 A047996

easy,nice,nonn

Wouter Meeussen (wouter.meeussen(AT)pandora.be)