A067059 -id:A067059 - cp's OEIS frontend

A084239 Rank of K-groups of Furstenberg transformation group C*-algebras of n-torus.

1, 2, 3, 4, 6, 8, 13, 20, 32, 52, 90, 152, 268, 472, 845, 1520, 2766, 5044, 9277, 17112, 31724, 59008, 110162, 206260, 387282, 729096, 1375654, 2601640, 4929378, 9358944, 17797100, 33904324, 64678112, 123580884, 236413054, 452902072

Offset: 0

Kamran Reihani (reyhan_k(AT)modares.ac.ir), Jun 21 2003

nonn

N. J. A. Sloane, Table of n, a(n) for n = 0..500
K. Reihani, C*-algebras from Anzai flows and their K-groups, arXiv preprint arXiv:math/0311425 [math.OA], 2003.
K. Reihani, K-theory of Furstenberg transformation group C^*-algebras, arXiv preprint arXiv:1109.4473 [math.OA], 2011.

Cf. A000980.

A084239 := proc(n)
    local tt,c ;
    if type(n,'odd') then
        product( 1+t^(i-(n+1)/2),i=1..n) ;
    else
        (1+t^(1/2))*product( 1+t^(i-(n+1)/2),i=1..n) ;
    end if;
    tt := expand(%) ;
    for c in tt do
        if c = lcoeff(c) then
            return c ;
        end if;
    end do:
end proc: # R. J. Mathar, Nov 13 2016

a[n_] := SeriesCoefficient[If[OddQ[n], 1, 1 + Sqrt[t]]*Product[1 + t^(i - (n + 1)/2), {i, n}], {t, 0, 0}];
Array[a, 36, 0] (* Jean-François Alcover, Nov 24 2017 *)

a(n) = constant term of prod(i=1, n, 1+t^(i-.5(n+1))) for odd n and a(n) = constant term of (1+t^(.5))*prod(i=1, n, 1+t^(i-.5(n+1))) for even n.

Sums of antidiagonals of A067059, i.e. a(n) is sum over k of number of partitions of [k(n-k)/2] into up to k parts each no more than n-k. Close to 2^(n+1)*sqrt(6/(Pi*n^3)) and seems to be even closer to something like 2^(n+1)*sqrt(6/(Pi*(n^3+0.9*n^2-0.1825*n+1.5))). - Henry Bottomley, Jul 20 2003

More terms from Henry Bottomley, Jul 20 2003

A241810 Number of balanced orbitals over n sectors.

Original entry on oeis.org

1, 1, 0, 0, 2, 6, 0, 6, 8, 36, 0, 88, 58, 376, 0, 1096, 526, 4476, 0, 14200, 5448, 57284, 0, 190206, 61108, 764812, 0, 2615268, 723354, 10499504, 0, 36677626, 8908546, 147110276, 0, 522288944, 113093022

Offset: 0

Peter Luschny, Apr 29 2014

For the combinatorial definitions see A232500. An orbital is balanced if its integral is 0. The integral of an orbital w over n sectors is Sum_{k=1..n} Sum_{i=1..k} w(i) where w(i) are the jumps of the orbital represented by -1, 0, 1.

Cf. A232500, A242087.

np[z_]:=Module[{i,j},For[i=Length[z],i>1&&z[[i-1]]>=z[[i]],i--];For[j=Length[z],z[[j]]<=z[[i-1]],j--];Join[Take[z,i-2],{z[[j]]},Reverse[Drop[ReplacePart[z,z[[i-1]],j],i-1]]]];o=Table[1,{16}];
n=0;f=0;Print[1];Print[1];While[n<16,n++;f=1-f;If[OddQ[f*n],Print[0],p=Join[-Take[o,n],{f},Take[o,n-f]];c=0;Do[If[Accumulate[Accumulate[p]][[-1]]==0,c++];p=np[p],{(2*n+1-f)!/(2*n!^2)}];Print[2*c]];n=n-f]
(* Hans Havermann, May 10 2014 *)

def A241810(n):
    if n == 0: return 1
    A = 0
    T = [0] if is_odd(n) else []
    for i in (1..n//2):
        T.append(-1); T.append(1)
    for p in Permutations(T):
        P = 0; S = 0
        for k in (0..n-1):
            P += p[k]; S += P
        if S == 0: A += 1
    return A
[A241810(n) for n in (0..32)]

a(2*n) = A204459(2, n).
a(2*n+1) = A242087(n).
a(4*n) = A063074(n) = A029895(2*n) = A067059(2*n, 2*n).
a(4*n+2) = 0 for all n (proved by H. Havermann).

More terms from Hans Havermann, May 10 2014

a(35), a(36) from Hans Havermann, May 23 2014

A069322 Square array read by antidiagonals of floor[(n+k)^(n+k)/(n^n*k^k)].

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 9, 16, 9, 1, 1, 12, 28, 28, 12, 1, 1, 14, 45, 64, 45, 14, 1, 1, 17, 65, 119, 119, 65, 17, 1, 1, 20, 89, 198, 256, 198, 89, 20, 1, 1, 23, 117, 307, 484, 484, 307, 117, 23, 1, 1, 25, 149, 449, 837, 1024, 837, 449, 149, 25, 1, 1, 28, 184, 629

Offset: 0

Henry Bottomley, Mar 14 2002

T(n,k)*sqrt(3)/(n*k*Pi) provides a rough approximation for A067059.

a(n,k) is an analog of the binomial coefficients over transformations instead of permutations. - Chad Brewbaker, Nov 25 2013

			Rows start: 1,1,1,1,1,1,...; 1,4,6,9,12,14,...; 1,6,16,28,45,65,...; 1,9,28,64,119,198,...; etc. T(3,5)=floor[8^8/(3^3*5^5)]=floor[16777216 /84375]=floor[198.84...]=198.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Initial columns and rows are A000012 and A060644, main diagonal is A000302.

t[n_, 0] := 1; t[n_, n_] := 1; t[n_, k_] := Floor[(n^n)/((k^k)*((n - k)^(n - k)))];  Table[t[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 22 2018 *)

for(n=0,15, for(k=0,n, print1(if(k==0, 1, if(k==n,1,floor((n^n)/(( k^k)*((n - k)^(n - k)))))), ", "))) \\ G. C. Greubel, Apr 22 2018

def transitorial(n)
    return n**n
end
def transnomial(n,k)
       return transitorial(n)/(transitorial(k) *transitorial(n-k))
end
0.upto(15) do |i|
   0.upto(i) do |j|
       print transnomial(i,j).to_s + " "
   end
   puts ""
end # Chad Brewbaker, Nov 25 2013

a(n,k) = (n^n) /((k^k)*((n-k)^(n-k))). - Chad Brewbaker, Nov 25 2013

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A084239 Rank of K-groups of Furstenberg transformation group C*-algebras of n-torus.

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A241810 Number of balanced orbitals over n sectors.

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A069322 Square array read by antidiagonals of floor[(n+k)^(n+k)/(n^n*k^k)].

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