A115361 -id:A115361 - cp's OEIS frontend

A129360 A054525 * A115361.

1, 0, 1, -1, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 10 2007

Row sums = A209229 (1, 1, 0, 1, 0, 0, 0, 1, ...).

A129353 = the inverse Möbius transform of A115361.

			First few rows of the triangle are:
   1;
   0,  1;
  -1,  0,  1;
   0,  0,  0,  1;
  -1,  0,  0,  0,  1;
   0, -1,  0,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   0,  0,  0,  0,  0,  0,  0,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A087003 (Moebius transform of A209229).
Row sums are A209229.
Cf. A054525, A115361, A129353, A001511, A103994, A129501.

tabl(nn) = {Tm = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); Tr = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = Tr^(-1); Tp = Tm*Ti; for (n=1, nn, for (k=1, n, print1(Tp[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

T(n, k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 03 2018

Moebius transform of A115361.

T(n,k) = A087003(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

More terms from Michel Marcus, Mar 28 2015

Offset changed by Andrew Howroyd, Aug 03 2018

A103994 A129360 * A115361.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 1, 1, 0, 1, -1, 0, 0, 0, 1, -1, -1, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 1, -1, 0, 1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 15 2007

Row sums = A104117: (1, 2, 0, 3, 0, 0, 0, 4, 0, 0, ...).

			First few rows of the triangle are:
   1;
   1,  1;
  -1,  0,  1;
   1,  1,  0,  1;
  -1,  0,  0,  0,  1;
  -1, -1,  1,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   1,  1,  0,  1,  0,  0,  0,  1;
   ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A209635 (Moebius transform of A104117).
Row sums are A104117.
Cf. A129360, A115361.

T[n_, k_] := If[Divisible[n, k], MoebiusMu[(n/k)/2^IntegerExponent[n/k, 2]], 0];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 14 2019 *)

tabl(nn) = {Tm = matrix(nn, nn, n, k, if (! (n % k), moebius(n/k), 0)); Tr = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); Ti = Tr^(-1); Tp = Tm*Ti*Ti; for (n=1, nn, for (k=1, n, print1(Tp[n, k], ", ");); print(););}

T(n, k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 03 2018

A129360 * A115361 as infinite lower triangular matrices.

T(n,k) = A209635(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

More terms from Michel Marcus, Mar 28 2015

A129501 A103994 * A115361.

Original entry on oeis.org

1, 2, 1, -1, 0, 1, 3, 2, 0, 1, -1, 0, 0, 0, 1, -2, -1, 2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -2, -1, 0, 0, 2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -3, -2, 3, -1, 0, 2, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 17 2007

Row sums = A129502: (1, 3, 0, 6, 0, 0, 0, 10, 0, 0, ...).

			First few rows of the triangle are:
   1;
   2,  1;
  -1,  0,  1;
   3,  2,  0,  1;
  -1,  0,  0,  0,  1;
  -2, -1,  2,  0,  0,  1;
  -1,  0,  0,  0,  0,  0,  1;
   4,  3,  0,  2,  0,  0,  0,  1;
   0,  0, -1,  0,  0,  0,  0,  0,  1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A317673 (Moebius transform of A129502).
Row sums are A129502.
Cf. A103994, A115361, A129360, A000217, A001511.

b[n_] := Module[{e}, Sum[e = IntegerExponent[d, 2]; If[d == 2^e, MoebiusMu[n/d] Binomial[2 + e, 2], 0], {d, Divisors[n]}]];
T[n_, k_] := If[Divisible[n, k], b[n/k], 0];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 24 2019 *)

T(n,k)={ if(n%k, 0, sumdiv(n/k, d, my(e=valuation(d, 2)); if(d==1<Andrew Howroyd, Aug 03 2018

A103994 * A115361 as infinite lower triangular matrices.

T(n,k) = A317673(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 03 2018

Terms a(56) and beyond from Andrew Howroyd, Aug 03 2018

A129353 A051731 * A115361.

Original entry on oeis.org

1, 2, 1, 1, 0, 1, 3, 2, 0, 1, 1, 0, 0, 0, 1, 2, 1, 2, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 4, 3, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 2, 3, 1, 0, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 10 2007

The inverse Moebius transform of the first column of A115361 which is A209229 gives the first column of this sequence.

			First few rows of the triangle are:
  1;
  2, 1;
  1, 0, 1;
  3, 2, 0, 1;
  1, 0, 0, 0, 1;
  2, 1, 2, 0, 0, 1;
  1, 0, 0, 0, 0, 0, 1;
  4, 3, 0, 2, 0, 0, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A001511.
Row sums are A129628 (inverse Moebius transform of A001511).
Cf. A051731, A115361.

A129353 := proc(n,k)
        add( A051731(n,j)*A115361(j-1,k-1),j=k..n) ;
end proc: # R. J. Mathar, Jul 14 2012

T[n_, k_] := If[Mod[n, k] != 0, 0, 1 + IntegerExponent[n/k, 2]];
Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 08 2020, from PARI *)

T(n, k)={if(n%k, 0, 1 + valuation(n/k,2))} \\ Andrew Howroyd, Aug 04 2018

T(n,k) = A001511(n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 04 2018

A129559 A054523 * A115361.

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 4, 2, 0, 1, 4, 0, 0, 0, 1, 4, 2, 2, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 8, 4, 0, 0, 2, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 8, 4, 4, 2, 0, 2, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Apr 20 2007

row sums = A129527: (1, 3, 3, 7, 5, 9, 7, 15, ...). Left column = phi(2*n), A062570: (1, 2, 2, 4, 4, 4, 6, 8, ...).

			First few rows of the triangle:
  1;
  2, 1;
  2, 0, 1;
  4, 2, 0, 1;
  4, 0, 0, 0, 1;
  4, 2, 2, 0, 0, 1;
  6, 0, 0, 0, 0, 0, 1;
  8, 4, 0, 2, 0, 0, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Column 1 is A062570.
Row sums are A129527 (inverse Moebius transform of A062570).
Cf. A054523, A115361.

T(n, k)=if(n%k, 0, eulerphi(2*n/k)) \\ Andrew Howroyd, Aug 07 2018

Equals A054523 * A115361 as infinite lower triangular matrices.

T(n,k) = phi(2*n/k) for k | n, T(n,k) = 0 otherwise. - Andrew Howroyd, Aug 07 2018

Terms a(56) and beyond from Andrew Howroyd, Aug 07 2018

A128807 A115361 * A007318.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, 3, 4, 3, 1, 1, 4, 6, 4, 1, 2, 7, 11, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 4, 11, 24, 36, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 2, 13, 42, 88, 127, 126, 84, 36, 9, 1

Offset: 0

Gary W. Adamson, Apr 06 2007

Row sums = A127804: (1, 3, 4, 11, 16, 36, 64, ...). Left column = the Ruler sequence, A001511: (1, 2, 1, 3, 1, 2, 1, 4, ...). A129262 = A007318 * A115361.

			First few rows of the triangle:
  1;
  2,  1;
  1,  2,  1;
  3,  4,  3,  1;
  1,  4,  6,  4,  1;
  2,  7, 11, 10,  5,  1;
  1,  6, 15, 20, 15,  6,  1;
  4, 11, 24, 36, 35, 21,  7,  1;
  ...

Cf. A115361, A007318, A129262, A001511, A127804.

A115361 * A007318 as infinite lower triangular matrices.

A129262 Triangle read by rows, A007318 * A115361.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 1, 9, 8, 6, 4, 1, 16, 15, 11, 10, 5, 1, 27, 26, 21, 20, 15, 6, 1, 44, 43, 42, 36, 35, 21, 7, 1, 73, 72, 84, 64, 70, 56, 28, 8, 1, 130, 129, 162, 120, 127, 126, 84, 36, 9, 1, 251, 250, 297, 240, 220, 252, 210, 120, 45, 10, 1, 507, 506, 518, 495, 385, 463, 462, 330, 165, 55, 11, 1

Offset: 1

Gary W. Adamson, Apr 06 2007

Row sums give A106461.
Left column is A119968.
A128807 = A115361 * A007318.

			First few rows of the triangle:
  1;
  2,  1;
  3,  2,  1;
  5,  4,  3,  1;
  9,  8,  6,  4,  1;
 16, 15, 11, 10,  5,  1;
 27, 26, 21, 20, 15,  6,  1;
 44, 43, 42, 36, 35, 21,  7,  1;
  ...

Cf. A007318, A115361, A106461, A119968, A128807.

Binomial transform of A115361.

Terms a(23) and following corrected by Georg Fischer, Jul 04 2023

A129264 Triangle read by rows: A000012 * A115361 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Apr 06 2007

Triangle read by rows, A000012 * A115361; where A000012 = an infinite lower triangular matrix with all 1's: [1; 1,1; 1,1,1; ...] and A115361 = the ruler function triangle.
The operation A000012 * A115361 takes partial sums of A115361 column terms starting from the top. [Gary W. Adamson, Nov 27 2009]
Eigensequence of the triangle = A089067: (1, 3, 5, 13, 23, 51, 97, 207, ...). [Gary W. Adamson, Nov 27 2009]
Row sums = A005187: (1, 3, 4, 7, 8, 10, 11, 15, ...).
Left column = A070939 starting (1, 2, 2, 3, 3, 3, 3, 4, 4, 4, ...).

			First few rows of the triangle:
  1;
  2, 1;
  2, 1, 1;
  3, 2, 1, 1;
  3, 2, 1, 1, 1;
  3, 2, 2, 1, 1, 1;
  3, 2, 2, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1;
  4, 3, 2, 2, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 2, 1, 1, 1, 1, 1;
  4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1;
  4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1;
  ...

Cf. A000012, A005187, A070939, A089067, A115361, A129265.

Incorrect a(79) removed by Georg Fischer, Jul 04 2023

A168261 Triangle read by rows, A115361 * the diagonalized variant of A018819.

Original entry on oeis.org

1, 1, 1, 0, 0, 2, 1, 1, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 6, 1, 1, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 4, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20

Offset: 1

Gary W. Adamson, Nov 21 2009

Row sums = A018819 starting with offset 1; (1, 2, 2, 4, 4, 6, 6, 10, 10,...).
Equals the eigensequence of triangle A115361.
Rightmost diagonal = A018819.
Sum of n-th row terms = rightmost term of next row.

			First few rows of the triangle =
1;
1, 1;
0, 0, 2;
1, 1, 0, 2;
0, 0, 0, 0, 4;
0, 0, 2, 0, 0, 4;
0, 0, 0, 0, 0 0, 6;
1, 1, 0, 2, 0, 0, 0, 6;
0, 0, 0, 0, 0, 0, 0, 0, 10;
0, 0, 0, 0, 4, 0, 0, 0, 0, 10;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14;
0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 14;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20;
0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 20;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26;
1, 1, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 26;
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 36;
...

Cf. A115361, A018819.

Equals M*Q as infinite lower triangular matrices, where M = triangle A115361, and Q = the diagonalized variant of A018819 such that (1, 1, 2, 2, 4, 4, 6, 6,...) = rightmost diagonal with the rest zeros.

A168312 Triangle read by rows, replace 1's of triangle A115361 starting from the right with (1, 2, 4, 8, ...).

Original entry on oeis.org

1, 2, 1, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Nov 22 2009

Row sums = A038712: (1, 3, 1, 7, 1, 3, 1, 15, ...).

			First few rows of the triangle:
   1;
   2, 1;
   0, 0, 1;
   4, 2, 0, 1;
   0, 0, 0, 0, 1;
   0, 0, 2, 0, 0, 1;
   0, 0, 0, 0, 0, 0, 1;
   8, 4, 0, 2, 0, 0, 0, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 1;
   0, 0, 0, 0, 2, 0, 0, 0, 0, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  16, 8, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1;
   0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Cf. A115361, A038712.

Triangle read by rows, replace 1's in triangle A115361 starting from the right with (1, 2, 4, 8, ...).

cp's OEIS Frontend

A129360 A054525 * A115361.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A103994 A129360 * A115361.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A129501 A103994 * A115361.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A129353 A051731 * A115361.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A129559 A054523 * A115361.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A128807 A115361 * A007318.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A129262 Triangle read by rows, A007318 * A115361.

Views

Author

Keywords

Comments

Examples