A293532 -id:A293532 - cp's OEIS frontend

A293530 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/Product_{j > 0, j mod k > 0} exp(x^j).

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, -1, -7, 0, 1, -1, -1, 5, 25, 0, 1, -1, -1, -1, -23, -181, 0, 1, -1, -1, -1, 25, -41, 1201, 0, 1, -1, -1, -1, 1, -101, 1111, -10291, 0, 1, -1, -1, -1, 1, 139, -209, -6259, 97777, 0, 1, -1, -1, -1, 1, 19, -569, 251, -16015

Offset: 0

Seiichi Manyama, Oct 11 2017

			Square array begins:
   1,    1,   1,    1,   1, ...
   0,   -1,  -1,   -1,  -1, ...
   0,    1,  -1,   -1,  -1, ...
   0,   -7,   5,   -1,  -1, ...
   0,   25, -23,   25,   1, ...
   0, -181, -41, -101, 139, ...

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

Columns k=1..3 give A000007, A293532, A293533.
Rows n=0 gives A000012.
Main diagonal gives A293116.
Cf. A293525.

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

A293565 Expansion of e.g.f. Product_{k>=0} exp(-x^(3*k+1)).

Original entry on oeis.org

1, -1, 1, -1, -23, 119, -359, -4201, 58801, -359857, -2121839, 74851919, -784376999, -1072414201, 192596540137, -3167640042841, 9241215940321, 860635956717599, -21290802661288799, 157132294452113183, 5980504635868174921, -220612220726623344361

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..447

E.g.f.: Product_{m>=0} exp(-x^(k*m+1)): A293116 (k=1), A293532 (k=2), this sequence (k=3), A293566 (k=4).

Cf. A293493.

CoefficientList[Series[E^(x/(x^3 - 1)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)

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

N=66; x='x+O('x^N); Vec(serlaplace(1/prod(m=0, N, exp(x^(3*m+1)))))

E.g.f.: exp(x/(x^3 - 1)).
a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k+1)! * a(n-3*k-1). - Ilya Gutkovskiy, Feb 24 2022
a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^(n-k) * binomial(n-2*k-1,k)/(n-3*k)!. - Seiichi Manyama, Jun 08 2024

A293566 E.g.f.: Product_{m>=0} exp(-x^(4*m+1)).

Original entry on oeis.org

1, -1, 1, -1, 1, -121, 721, -2521, 6721, -378001, 5473441, -39972241, 199679041, -7005552841, 176899522801, -2186722497961, 17454339826561, -459473703430561, 16503993702423361, -306140370496394401, 3555223271216311681, -80917223353652470681

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

Muniru A Asiru, Table of n, a(n) for n = 0..50

E.g.f.: Product_{m>=0} exp(-x^(k*m+1)): A293116 (k=1), A293532 (k=2), A293565 (k=3), this sequence (k=4).

Cf. A293507.

seq(factorial(k) * coeftayl(product(exp(-x^(4*m + 1)),m = 0..k), x = 0, k),k = 0..50); # Muniru A Asiru, Oct 15 2017

CoefficientList[Series[E^(x/(x^4 - 1)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 13 2017 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(x/(x^4-1))))

N=66; x='x+O('x^N); Vec(serlaplace(1/prod(m=0, N, exp(x^(4*m+1)))))

E.g.f.: exp(x/(x^4 - 1)).

a(0) = 1; a(n) = -Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * (4*k+1)! * a(n-4*k-1). - Ilya Gutkovskiy, Feb 24 2022

A293567 Expansion of e.g.f.: exp(x^2/(x^3 - 1)).

Original entry on oeis.org

1, 0, -2, 0, 12, -120, -120, 5040, -38640, -181440, 5412960, -33264000, -478336320, 12194582400, -50871300480, -2168559993600, 49692144902400, -59775248332800, -15819216007795200, 329479616206540800, 1101564635255884800, -174845824790757120000

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

E.g.f.: Product_{k>0} exp(x^(-(m*k-1))): A293532 (m=2), this sequence (m=3), A293568 (m=4).

Cf. A293494.

With[{nn=30},CoefficientList[Series[Exp[x^2/(x^3-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 31 2024 *)

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

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

E.g.f.: Product_{k>0} exp(x^(-(3*k-1))).

Definition clarified by Harvey P. Dale, Mar 31 2024

A293568 E.g.f.: exp(x^3/(x^4 - 1)).

Original entry on oeis.org

1, 0, 0, -6, 0, 0, 360, -5040, 0, -60480, 3628800, -39916800, 19958400, -3113510400, 130767436800, -1318571654400, 3487131648000, -355687428096000, 12813639597158400, -126713646259200000, 1013709170073600000, -85161707377883136000, 2819368492175499264000

Offset: 0

Seiichi Manyama, Oct 12 2017

sign

E.g.f.: Product_{k>0} exp(x^(-(m*k-1))): A293532 (m=2), A293567 (m=3), this sequence (m=4).

Cf. A293526.

With[{nn=30},CoefficientList[Series[Exp[x^3/(x^4-1)],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 31 2020 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(x^3/(x^4-1))))

N=66; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, exp(x^(4*k-1)))))

E.g.f.: Product_{k>0} exp(x^(-(4*k-1))).

cp's OEIS Frontend

A293530 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/Product_{j > 0, j mod k > 0} exp(x^j).

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A293565 Expansion of e.g.f. Product_{k>=0} exp(-x^(3*k+1)).

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A293566 E.g.f.: Product_{m>=0} exp(-x^(4*m+1)).

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A293567 Expansion of e.g.f.: exp(x^2/(x^3 - 1)).

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A293568 E.g.f.: exp(x^3/(x^4 - 1)).

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