A204243 -id:A204243 - cp's OEIS frontend

A204444 Permanent of the n-th principal submatrix of A204443.

1, 1, 2, 9, 34, 140, 672, 4752, 32304, 230832, 1780704, 17927568, 175473216, 1780206336, 18988867584, 248341303296, 3173837230080, 41713375334400, 568818754560000, 9150176586240000, 144343174487040000, 2329812142295040000

Offset: 1

Clark Kimberling, Jan 15 2012

nonn

Cf. A204243, A204435.

f[i_, j_] := Floor[(i + j + 3)/4] - Floor[(i + j)/4];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 14}, {i, 1, n}]]   (* A204443 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 22}]    (* A204444 *)

A204242 Infinite symmetric matrix given by f(i,1)=1, f(1,j)=1, f(i,i)=2^i-1 and f(i,j)=0 otherwise, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 0, 0, 1, 1, 0, 7, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 15, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 31, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 63, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 127, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0

Offset: 1

Clark Kimberling, Jan 13 2012

			Northwest corner:
1 1 1 1
1 3 0 0
1 0 7 0
1 0 0 15

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A204243, A203011.

N:= 1000: # to get a(1) to a(N)
V:= Vector(N):
V[[seq(k*(k+1)/2, k= 1..floor((sqrt(8*N+1)-1)/2))]]:= 1:
V[[seq(1+k*(k+1)/2, k=1..floor((sqrt(8*N-7)-1)/2))]]:= 1:
V[[seq(1+2*k+2*k^2, k=0..floor((sqrt(2*N-1)-1)/2))]]:=
    :
convert(V,list); # Robert Israel, Nov 30 2015

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := 2^i - 1;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]     (* A204242 *)
Table[Det[m[n]], {n, 1, 15}]  (* A204243 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 15}]   (* A203011 *)

From Robert Israel, Nov 30 2015: (Start)
a(k*(k+1)/2) = a(1 + k*(k+1)/2) = 1.
a(2*k^2 + 2*k + 1) = 2^(k+1) - 1.
a(n) = 0 otherwise. (End)

Name edited by Robert Israel, Nov 30 2015

A377278 Denominators in a harmonic triangle; q-analog of A126615, here q = 2.

Original entry on oeis.org

1, 3, 3, 3, 21, 7, 3, 21, 105, 15, 3, 21, 105, 465, 31, 3, 21, 105, 465, 1953, 63, 3, 21, 105, 465, 1953, 8001, 127, 3, 21, 105, 465, 1953, 8001, 32385, 255, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 511, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 522753, 1023

Offset: 1

Werner Schulte, Oct 22 2024

The harmonic triangle uses the terms of this sequence as denominators, numerators = 1. The inverse of the harmonic triangle has entries -2^(n-k-1) for 1<=k

Conjecture: Row sums of the harmonic triangle are A204243(n) / A005329(n).

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :  1   2    3    4     5     6      7       8       9    10
================================================================
   1 :  1
   2 :  3   3
   3 :  3  21    7
   4 :  3  21  105   15
   5 :  3  21  105  465    31
   6 :  3  21  105  465  1953    63
   7 :  3  21  105  465  1953  8001    127
   8 :  3  21  105  465  1953  8001  32385     255
   9 :  3  21  105  465  1953  8001  32385  130305     511
  10 :  3  21  105  465  1953  8001  32385  130305  522753  1023
  etc.
The harmonic triangle starts:
  [1]  1/1
  [2]  1/3   1/3
  [3]  1/3  1/21    1/7
  [4]  1/3  1/21  1/105   1/15
  [5]  1/3  1/21  1/105  1/465    1/31
  [6]  1/3  1/21  1/105  1/465  1/1953  1/63
  etc.
The inverse of the harmonic triangle starts:
  [1]    1
  [2]   -1   3
  [3]   -2  -1   7
  [4]   -4  -2  -1  15
  [5]   -8  -4  -2  -1  31
  [6]  -16  -8  -4  -2  -1  63
  etc.

Cf. A000225, A005329, A006095, A126615, A204243.

PARI
```
T(n,k)=if(k
				
```

T(n, k) = (2^k - 1) * (2^(k+1) - 1) for 1 <= k < n; T(n, n) = 2^n - 1.
Sum_{k=1..n} 2^(k-1) / T(n, k) = 1.
Product_{k=1..n} T(n, k)^((-1)^k) = 1.
Row sums are n + 4 * (2^n - 1) * (2^(n-1) - 1) / 3 = n + 4 * A006095(n).
G.f.: x*y*(1 + 2*x - 4*x*y + 4*x^2*y)/((1 - x)*(1 - x*y)(1 - 2*x*y)*(1 - 4*x*y)). - Stefano Spezia, Oct 23 2024

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A204444 Permanent of the n-th principal submatrix of A204443.

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A204242 Infinite symmetric matrix given by f(i,1)=1, f(1,j)=1, f(i,i)=2^i-1 and f(i,j)=0 otherwise, read by antidiagonals.

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A377278 Denominators in a harmonic triangle; q-analog of A126615, here q = 2.

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