A051731 -id:A051731 - cp's OEIS frontend

A101508 Product of binomial matrix and the Mobius matrix A051731.

1, 2, 1, 4, 2, 1, 8, 4, 3, 1, 16, 8, 6, 4, 1, 32, 16, 11, 10, 5, 1, 64, 32, 21, 20, 15, 6, 1, 128, 64, 42, 36, 35, 21, 7, 1, 256, 128, 85, 64, 70, 56, 28, 8, 1, 512, 256, 171, 120, 127, 126, 84, 36, 9, 1, 1024, 512, 342, 240, 220, 252, 210, 120, 45, 10, 1, 2048, 1024, 683, 496, 385, 463, 462, 330, 165, 55, 11, 1

Offset: 0

Paul Barry, Dec 05 2004

Row sums are A101509. Diagonal sums are A101510.
The matrix inverse appears to be A128313. - R. J. Mathar, Mar 22 2013
Read as upper triangular matrix, this can be seen as "recurrences in A135356 applied to A023531" [Paul Curtz, Mar 03 2017]. - The columns are: A000079, A131577, A024495, A000749, A139761, ... Column n differs after the (n+1)-th nonzero term on from the binomial coefficients C(k,n). - M. F. Hasler, Mar 05 2017

			Rows begin
  1;
  2,1;
  4,2,1;
  8,4,3,1;
  16,8,6,4,1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

A101508 := proc(n,k)
    a := 0 ;
    for i from 0 to n do
        if modp(i+1,k+1) = 0 then
            a := a+binomial(n,i) ;
        end if;
    end do:
    return a;
end proc: # R. J. Mathar, Mar 22 2013

t[n_, k_] := Sum[If[Mod[i + 1, k + 1] == 0, Binomial[n, i], 0], {i, 0, n}]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2014 *)

T(n,k)=sum(i=0,n, if((i+1)%(k+1)==0, binomial(n, i))) \\ M. F. Hasler, Mar 05 2017

T(n, k) = Sum_{i=0..n} if(mod(i+1, k+1)=0, binomial(n, i), 0).

Rows have g.f. x^k/((1-x)^(k+1)-x^(k+1)).

A113998 Reverse of triangle A051731.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1

Offset: 1

Paul Barry, Nov 12 2005

Row sums are A000005. Diagonal sums are A001227 (?). Row k corresponds to A113999(k).

			Triangle begins
1;
1,1;
1,0,1;
1,0,1,1;
1,0,0,0,1;
1,0,0,1,1,1;
1,0,0,0,0,0,1;
1,0,0,0,1,0,1,1;

Sum_{k, 1<=k<=n} k*T(n,k)=A081307(n) - Philippe Deléham, Feb 03 2007

A127446 Triangle T(n,k) = n*A051731(n,k) read by rows.

Original entry on oeis.org

1, 2, 2, 3, 0, 3, 4, 4, 0, 4, 5, 0, 0, 0, 5, 6, 6, 6, 0, 0, 6, 7, 0, 0, 0, 0, 0, 7, 8, 8, 0, 8, 0, 0, 0, 8, 9, 0, 9, 0, 0, 0, 0, 0, 9, 10, 10, 0, 0, 10, 0, 0, 0, 0, 10, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 12, 12, 12, 12, 0, 12, 0, 0, 0, 0, 0, 12, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, 14, 0, 0, 0, 0, 14

Offset: 1

Gary W. Adamson, Jan 14 2007

Replace the 1's in row n of A051731 with n's.

T(n,k) is the sum of the k's in the partitions of n into equal parts. - Omar E. Pol, Nov 25 2019

			First few rows of the triangle:
  1;
  2, 2;
  3, 0, 3;
  4, 4, 0, 4;
  5, 0, 0, 0, 5;
  6, 6, 6, 0, 0, 6;
  7, 0, 0, 0, 0, 0, 7;
  ...
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the sum of the k's are [6, 6, 6, 0, 0, 6] respectively, equaling the 6th row of triangle. - _Omar E. Pol_, Nov 25 2019

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A038040 (row sums), A051731, A126988, A244051, A328362.

a127446 n k = a127446_tabl !! (n-1) !! (k-1)
a127446_row n = a127446_tabl !! (n-1)
a127446_tabl = zipWith (\v ws -> map (* v) ws) [1..] a051731_tabl
-- Reinhard Zumkeller, Jan 21 2014

A127446 := proc(n,k) if n mod k = 0 then n; else 0; fi; end: for n from 1 to 20 do for k from 1 to n do printf("%d,",A127446(n,k)) ; od: od: # R. J. Mathar, May 08 2009

Flatten[Table[If[Mod[n, k] == 0, n, 0], {n, 20}, {k, n}]] (* Vincenzo Librandi, Nov 02 2016 *)

T(n,k) = k*A126988(n,k). - Omar E. Pol, Nov 25 2019

Edited and extended by R. J. Mathar, May 08 2009

A127639 A051731 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

Original entry on oeis.org

2, 2, 3, 2, 0, 5, 2, 3, 0, 7, 2, 0, 0, 0, 11, 2, 3, 5, 0, 0, 13, 2, 0, 0, 0, 0, 0, 17, 2, 3, 0, 7, 0, 0, 0, 19, 2, 0, 5, 0, 0, 0, 0, 0, 23, 2, 3, 0, 0, 11, 0, 0, 0, 0, 29, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 2, 3, 5, 7, 0, 13, 0, 0, 0, 0, 0, 37, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 2, 3, 0, 0, 0, 0, 17, 0, 0

Offset: 1

Gary W. Adamson, Jan 21 2007

Row sums = A007445, inverse Mobius transform of the primes: (2, 5, 7, 12, 13, 23, ...)

			First few rows of the triangle are:
2;
2, 3;
2, 0, 5;
2, 3, 0, 7;
2, 0, 0, 0, 11;
2, 3, 5, 0, 0, 13;
...

Cf. A051731, A007445, A127639.

A051731 := proc(n,k) if n mod k = 0 then 1 ; else 0 ; fi ; end: A127639 := proc(n,k) A051731(n,k)*ithprime(k) ; end: for n from 1 to 16 do for k from 1 to n do printf("%d,", A127639(n,k)) ; od ; od ; # R. J. Mathar, Mar 14 2007

More terms from R. J. Mathar, Mar 14 2007

A127172 Cube of A051731.

Original entry on oeis.org

1, 3, 1, 3, 0, 1, 6, 3, 0, 1, 3, 0, 0, 0, 1, 9, 3, 3, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 10, 6, 0, 3, 0, 0, 0, 1, 6, 0, 3, 0, 0, 0, 0, 0, 1, 9, 3, 0, 0, 0, 3, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jan 06 2007

Nonzero terms in every column = A007425: (1, 3, 3, 6, 3, 9, 3, ...).
Row sums = A007426: (1, 4, 4, 20, 4, 16, ...).
A127172 * mu(n) = d(n); or A127172 * A008683 = A000005.
A127172 * d(n) = tau_5(n); or A127172 * A000005 = A061200.
A127172 * phi(n) = A007429: (1, 4, 5, 11, 7, 20, ...); or: A127172 * A000010 = A007429.
Note that A051731 * d(n) = row sums of A127172; or A051731 * A000005 = A007425.
Also, A126988 * mu(n) = phi(n); or A126988 * A008683 = A000010.
A126988 * phi(n) = A018804: (1, 3, 5, 8, 9, 15, ...); = A127170 * mu(n).

			First few rows of the triangle:
   1;
   3, 1;
   3, 0, 1;
   6, 3, 0, 1;
   3, 0, 0, 0, 1;
   9, 3, 3, 0, 0, 1;
   3, 0, 0, 0, 0, 0, 1;
  10, 6, 0, 3, 0, 0, 0, 1;
   6, 0, 3, 0, 0, 0, 0, 0, 1;
   9, 3, 0, 0, 3, 0, 0, 0, 0, 1;
  ...

Cf. A000005, A007425, A127170, A051731, A007429, A000010, A126988, A018804.

Cube of A051731 A007425: (1, 3, 3, 6, 3, 9, 3, ...) in every column k, interspersed with (k-1) zeros.

A128408 Triangle read by rows: A128407 * A051731 as infinite lower triangular matrices.

Original entry on oeis.org

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

Offset: 1

Gary W. Adamson, Mar 01 2007

Left and right borders = mu(n), A008683. Row sums = A008966: (1, -2, -2, 0, -2, 4, -2, 0, 0, 4, ...). A128408 * [1,2,3,...] = A063441: (1, -3, -4, 0, -6, 12, ...). A054524 = A051731 * A128407.

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

Cf. A008683, A008966, A054524, A063441, A128407.

a(45) = 0 inserted and more terms from Georg Fischer, Jun 05 2023

A133700 A051731 * A001227; a(n) = Sum_{d|n} A001227(d).

Original entry on oeis.org

1, 2, 3, 3, 3, 6, 3, 4, 6, 6, 3, 9, 3, 6, 9, 5, 3, 12, 3, 9, 9, 6, 3, 12, 6, 6, 10, 9, 3, 18, 3, 6, 9, 6, 9, 18, 3, 6, 9, 12, 3, 18, 3, 9, 18, 6, 3, 15, 6, 12, 9, 9, 3, 20, 9, 12, 9, 6, 3, 27, 3, 6, 18, 7, 9, 18, 3, 9, 9, 18, 3, 24, 3, 6, 18, 9, 9, 18, 3, 15, 15, 6, 3, 27, 9, 6, 9, 12, 3, 36, 9, 9, 9

Offset: 1

Gary W. Adamson, Sep 21 2007

			a(4) = sum of row 4 terms of triangle A133699: (1 + 1 + 0 + 1) = (1, 1, 0, 1) dot (1, 1, 2, 1), where A001227 = (1, 1, 2, 1, 2, 2, 2, 1, 3, ...).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A001227, A051731, A133699.

Cf. A001620, A082633.

f[p_, e_] := (e+1)*(e+2)/2; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 28 2023 *)

A133700(n) = sumdiv(n,d,numdiv(d>>valuation(d,2))); \\ Antti Karttunen, Sep 27 2018

Inverse Möbius transform of A001227, the number of odd divisors of n. Row sums of triangle A133699.
Dirichlet g.f. (zeta(s))^3*(1-1/2^s). - R. J. Mathar, Feb 07 2011
a(n) = Sum_{d|n} A001227(d). - Antti Karttunen, Sep 27 2018
Sum_{k=1..n} a(k) ~ n/4 * (log(n)^2 + (6*g - 2 + 2*log(2))*log(n) + 2 + 6*g^2 - log(2)^2 - 2*log(2) + 6*g*(log(2) - 1) - 6*sg1), where g is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant A082633. - Vaclav Kotesovec, Feb 02 2019
G.f.: Sum_{k>=1} tau(k)*x^k/(1 - x^(2*k)), where tau = A000005. - Ilya Gutkovskiy, Sep 13 2019
Multiplicative with a(2^e) = e+1, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Oct 28 2023

More terms from R. J. Mathar, Jan 19 2009

Second, equivalent formula added to the definition by Antti Karttunen, Sep 27 2018

A158902 Triangle read by rows: the matrix product A051731 * A158821.

Original entry on oeis.org

1, 2, 1, 3, 0, 1, 5, 1, 0, 1, 5, 0, 0, 0, 1, 9, 1, 1, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 12, 1, 0, 1, 0, 0, 0, 1, 11, 0, 1, 0, 0, 0, 0, 0, 1, 15, 1, 0, 0, 1, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson and Mats Granvik, Mar 29 2009

			First few rows of the triangle =
1;
2, 1;
3, 0, 1;
5, 1, 0, 1;
5, 0, 0, 0, 1;
9, 1, 1, 0, 0, 1;
7, 0, 0, 0, 0, 0, 1;
12, 1, 0, 1, 0, 0, 0, 1
11, 0, 1, 0, 0, 0, 0, 0, 1;
15, 1, 0, 0, 1, 0, 0, 0, 0, 1;
11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
23, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1;
13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
21, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
...

Cf. A158821, A051731, A158901, A000203 (row sums).

A158902 := proc(n,k)
    add( A051731(n,j)*A158821(j-1,k-1),j=k..n) ;
end proc:
seq(seq(A158902(n,k),k=1..n),n=1..12) ; # R. J. Mathar, Jan 08 2015

m = 12; (* number of rows *)
T1[n_, k_] := Boole[Mod[n, k] == 0];
T2[n_, k_] := Which[n == k, 1, k == 1, n-1, True, 0];
T = Array[T1, {m, m}].Array[T2, {m, m}];
Table[T[[n, k]], {n, m}, {k, n}] // Flatten (* Jean-François Alcover, Feb 01 2023 *)

T(n,n) = 1.

T(n,1) = A158901(n).

Wrong A-number in definition corrected by Robert Israel, Jan 08 2015

A176890 Triangle T(n,k) read by rows. Signed subsequence of A051731.

Original entry on oeis.org

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

Offset: 1

Mats Granvik and Gary W. Adamson, Apr 28 2010

Let A=A176890*A176890, B=A*A, C=B*B, D=C*C and so on, then the leftmost column in the last matrix converges to the Moebius function A008683.

			Table begins:
-1,
1,0,
1,0,0,
1,1,0,0,
1,0,0,0,0,
1,1,1,0,0,0,
1,0,0,0,0,0,0,
1,1,0,1,0,0,0,0,
1,0,1,0,0,0,0,0,0,
1,1,0,0,1,0,0,0,0,0,

Cf. A051731, A008683.

This is A176918 * the diagonalized mu(n).

=if(and(row()=1;column()=1);-1;if(mod(row();column())=0;1;0)-if(and(column()>1;row()=column());1;0))

T(n,k) = if n=1 and k=1 then -1 elseif n=k then 0 elseif k divides n then 1 else 0.

A127641 A127640 * A051731 as infinite lower triangular matrices.

Original entry on oeis.org

2, 3, 3, 5, 0, 5, 7, 7, 0, 7, 11, 0, 0, 0, 11, 13, 13, 13, 0, 0, 13, 17, 0, 0, 0, 0, 0, 17, 19, 19, 0, 19, 0, 0, 0, 19, 23, 0, 23, 0, 0, 0, 0, 0, 23, 29, 29, 0, 0, 29, 0, 0, 0, 0, 29, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 31, 37, 37, 37, 37, 0, 37, 0, 0, 0, 0, 0, 37, 41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41

Offset: 1

Gary W. Adamson, Jan 21 2007

A prime transform of A051731.

			First few rows of the triangle are:
2;
3, 3;
5, 0, 5;
7, 7, 0, 7;
11, 0, 0, 0, 11;
13, 13, 13, 0, 0, 13;
...

Cf. A127640, A051731, A127638, A127639.

A127640 := proc(n,m) if m < n then 0; else ithprime(n) ; fi ; end: A051731 := proc(n,k) if n mod k = 0 then 1 ; else 0 ; fi ; end: A127641 := proc(n,m) add( A127640(n,k)*A051731(k,m),k=1..n) ; end: for n from 1 to 15 do for m from 1 to n do printf("%d,",A127641(n,m)) ; od ; od ; # R. J. Mathar, May 19 2007

More terms from R. J. Mathar, May 19 2007

cp's OEIS Frontend

A101508 Product of binomial matrix and the Mobius matrix A051731.

Views

Author

Keywords

Comments

Examples

Links

Programs

Maple

Mathematica

PARI

Formula

A113998 Reverse of triangle A051731.

Views

Author

Keywords

Comments

Examples

Formula

A127446 Triangle T(n,k) = n*A051731(n,k) read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A127639 A051731 * A127640, where A127640 = infinite lower triangular matrix with the sequence of primes in the main diagonal and the rest zeros.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A127172 Cube of A051731.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A128408 Triangle read by rows: A128407 * A051731 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A133700 A051731 * A001227; a(n) = Sum_{d|n} A001227(d).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A158902 Triangle read by rows: the matrix product A051731 * A158821.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple