A038063 -id:A038063 - cp's OEIS frontend

A383011 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} mu(n/d) * (-k)^d.

1, 2, -1, 3, -3, 0, 4, -6, 2, 0, 5, -10, 8, -3, 0, 6, -15, 20, -18, 6, 0, 7, -21, 40, -60, 48, -11, 0, 8, -28, 70, -150, 204, -124, 18, 0, 9, -36, 112, -315, 624, -690, 312, -30, 0, 10, -45, 168, -588, 1554, -2620, 2340, -810, 56, 0, 11, -55, 240, -1008, 3360, -7805, 11160, -8160, 2184, -105, 0

Offset: 1

Seiichi Manyama, Apr 12 2025

			Square array begins:
   1,   2,    3,    4,     5,     6,      7, ...
  -1,  -3,   -6,  -10,   -15,   -21,    -28, ...
   0,   2,    8,   20,    40,    70,    112, ...
   0,  -3,  -18,  -60,  -150,  -315,   -588, ...
   0,   6,   48,  204,   624,  1554,   3360, ...
   0, -11, -124, -690, -2620, -7805, -19656, ...
   0,  18,  312, 2340, 11160, 39990, 117648, ...

Columns k=1..5 give A154955, A038063, A038064, A038065, A038066.
Main diagonal gives A383012.
Cf. A008683, A074650.

a(n, k) = -sumdiv(n, d, moebius(n/d)*(-k)^d)/n;

G.f. of column k: Sum_{j>=1} mu(j) * log(1 + k*x^j) / j.

Product_{n>=1} 1/(1 - x^n)^A(n,k) = 1 + k*x.

A320783 Inverse Euler transform of (-1)^(n - 1).

Original entry on oeis.org

1, 1, -2, 2, -3, 6, -11, 18, -30, 56, -105, 186, -335, 630, -1179, 2182, -4080, 7710, -14588, 27594, -52377, 99858, -190743, 364722, -698870, 1342176, -2581425, 4971008, -9586395, 18512790, -35792449, 69273666, -134215680, 260300986, -505294125, 981706806

Offset: 0

Gus Wiseman, Oct 22 2018

sign

After a(1) and a(2), same as A038063.

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

OEIS Wiki, Euler transform

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320778, A320779, A320780, A320781, A320782.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[(-1)^(#-1)&,30]]

cp's OEIS Frontend

A383011 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} mu(n/d) * (-k)^d.

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A320783 Inverse Euler transform of (-1)^(n - 1).

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