A293530 -id:A293530 - cp's OEIS frontend

A293532 E.g.f.: exp(x/(x^2 - 1)).

1, -1, 1, -7, 25, -181, 1201, -10291, 97777, -1013545, 12202561, -151573951, 2173233481, -31758579997, 524057015665, -8838296029291, 164416415570401, -3145357419120721, 65057767274601217, -1391243470549894135, 31671795881695430521, -747996624368605997701

Offset: 0

Seiichi Manyama, Oct 11 2017

sign

Column k=2 of A293530.

Cf. A088009.

CoefficientList[Series[E^(x/(x^2 - 1)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 12 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(x^2-1))))

E.g.f.: Product_{k>=1} 1/(1 + x^k)^(phi(k)/k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, May 25 2019

A293525 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. Product_{j > 0, j mod k > 0} exp(x^j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 7, 0, 1, 1, 3, 7, 25, 0, 1, 1, 3, 13, 49, 181, 0, 1, 1, 3, 13, 49, 321, 1201, 0, 1, 1, 3, 13, 73, 381, 2131, 10291, 0, 1, 1, 3, 13, 73, 381, 2971, 19783, 97777, 0, 1, 1, 3, 13, 73, 501, 3331, 26713, 195777, 1013545, 0, 1, 1

Offset: 0

Seiichi Manyama, Oct 11 2017

			Square array begins:
   1,   1,   1,   1,   1, ...
   0,   1,   1,   1,   1, ...
   0,   1,   3,   3,   3, ...
   0,   7,   7,  13,  13, ...
   0,  25,  49,  49,  73, ...
   0, 181, 321, 381, 381, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=1..3 give A000007, A088009, A113775.
Rows n=0 gives A000012.
Main diagonal gives A000262.
Cf. A293530.

kmax = 12; col[k_] := PadRight[(Exp[Sum[x^j, {j, 1, k - 1}]/(1 - x^k)] + O[x]^kmax // CoefficientList[#, x] &), kmax]*Range[0, kmax - 1]!; A = Array[col, kmax]; Table[A[[n - k + 1, k]], {n, 1, kmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 12 2017, from formula *)

E.g.f. of column k: exp((Sum_{j=1..k-1} x^j)/(1 - x^k)).

A293533 E.g.f.: 1/Product_{m > 0, m mod 3 > 0} exp(x^m).

Original entry on oeis.org

1, -1, -1, 5, -23, -41, 1111, -6259, -16015, 828143, -6453809, -23557931, 1516982809, -15821700025, -76745280793, 5613303472349, -73951449390239, -445513157340449, 36776986711862815, -582726291386478427, -4158268555818657079, 388618610293537423799

Offset: 0

Seiichi Manyama, Oct 11 2017

sign

Column k=3 of A293530.

CoefficientList[Series[E^(x*(1 + x)/(x^3 - 1)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 12 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp((x+x^2)/(x^3-1))))

N=66; x='x+O('x^N); Vec(serlaplace(prod(m=1, N, exp(x^(3*m))/exp(x^m))))

E.g.f.: exp((x + x^2)/(x^3 - 1)).

cp's OEIS Frontend

A293532 E.g.f.: exp(x/(x^2 - 1)).

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A293525 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. Product_{j > 0, j mod k > 0} exp(x^j).

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A293533 E.g.f.: 1/Product_{m > 0, m mod 3 > 0} exp(x^m).

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Mathematica

PARI

PARI

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