A114004 -id:A114004 - cp's OEIS frontend

A114006 Row sums of number triangle A114004.

1, -1, -1, 1, -1, 3, -1, -1, 1, 3, -1, -5, -1, 3, 3, 1, -1, -5, -1, -5, 3, 3, -1, 7, 1, 3, -1, -5, -1, -13, -1, -1, 3, 3, 3, 13, -1, 3, 3, 7, -1, -13, -1, -5, -5, 3, -1, -9, 1, -5, 3, -5, -1, 7, 3, 7, 3, 3, -1, 31, -1, 3, -5, 1, 3, -13, -1, -5, 3, -13, -1, -25, -1, 3, -5, -5, 3, -13, -1, -9, 1, 3, -1, 31, 3, 3, 3, 7, -1, 31, 3, -5, 3, 3, 3, 11

Offset: 1

Paul Barry, Nov 12 2005

A signed version of A050328.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A008836, A008966, A050328, A114004, A114005.

a050328[n_]:=If[n==1, n, Sum[If[(dIndranil Ghosh, May 27 2017 *)

A050328(n) = if(1==n,n,sumdiv(n,d,if((dA050328(d),0)));
A114006(n) = ((-1)^bigomega(n) * A050328(n)); \\ Antti Karttunen, May 27 2017

a(n) = A008836(n) * A050328(n). - Antti Karttunen, May 27 2017

a(1) = 1; a(n) = Sum_{d|n, d < n} mu(n/d) * a(d). - Ilya Gutkovskiy, Feb 23 2020

More terms, incorrect formulas replaced by a correct one - Antti Karttunen, May 27 2017

A114005 First column of number triangle A114004.

Original entry on oeis.org

1, -2, -2, 2, -2, 6, -2, -2, 2, 6, -2, -10, -2, 6, 6, 2, -2, -10, -2, -10, 6, 6, -2, 14, 2, 6, -2, -10, -2, -26, -2, -2, 6, 6, 6, 26, -2, 6, 6, 14, -2, -26, -2, -10, -10, 6, -2, -18, 2, -10, 6, -10, -2, 14, 6, 14, 6, 6, -2, 62, -2, 6, -10, 2, 6, -26, -2, -10, 6, -26, -2, -50, -2, 6, -10, -10, 6, -26, -2, -18, 2, 6, -2, 62, 6, 6, 6, 14, -2, 62, 6, -10, 6, 6, 6

Offset: 1

Paul Barry, Nov 12 2005

Moebius transform of A114006. - Mats Granvik, Jan 02 2009

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A050328, A114006.

A050328(n) = if(1==n,n,sumdiv(n,d,if((dA050328(d),0)));
A114006(n) = ((-1)^bigomega(n) * A050328(n));
A114005(n) = if(1==n,n,2*A114006(n)); \\ Antti Karttunen, May 27 2017

a(1) = 1, for n > 1, a(n) = 2*A114006(n). - Corrected by Antti Karttunen, May 27 2017

a(1) = 1; a(n) = -2 * Sum_{d|n, d < n} a(d). - Ilya Gutkovskiy, Feb 23 2020

More terms from Antti Karttunen, May 27 2017

A114002 Expansion of g.f. x^k(1+x^(k+1))/(1-x^(k+1)).

Original entry on oeis.org

1, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 0, 0, 0, 1, 2, 2, 2, 0, 0, 1, 2, 0, 0, 0, 0, 0, 1, 2, 2, 0, 2, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 0, 0, 1, 2, 2, 0, 0, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Paul Barry, Nov 12 2005

Inverse is A114004. Row sums are A114003.

			Triangle begins:
  1;
  2, 1;
  2, 0, 1;
  2, 2, 0, 1;
  2, 0, 0, 0, 1;
  2, 2, 2, 0, 0, 1;
  2, 0, 0, 0, 0, 0, 1;
  ...

Cf. A051731, A114003, A114004.

T[n_,k_]:=SeriesCoefficient[x^k(1+x^(k+1))/(1-x^(k+1)),{x,0,n}]; Table[T[n,k],{n,0,13},{k,0,n}] //Flatten (* Stefano Spezia, Sep 08 2023 *)

Column k has g.f. x^k(1+x^(k+1))/(1-x^(k+1)).

Equals 2*A051731 - I, I = Identity matrix. - Gary W. Adamson, Nov 07 2007

cp's OEIS Frontend

A114006 Row sums of number triangle A114004.

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A114005 First column of number triangle A114004.

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A114002 Expansion of g.f. x^k(1+x^(k+1))/(1-x^(k+1)).

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