A293236 -id:A293236 - cp's OEIS frontend

A293267 Coefficients in asymptotic expansion of sequence A293236.

1, 0, -1, 0, 3, 12, 35, 61, -153, -2197, -11330, -6513, 576693, 8039161, 74592474, 536729860, 2823657417, 5689440944, -93424479237, -1452915372079, -9514285768034, 42246743764900, 2340471395600501, 42987651613157615, 564523206912685856, 5656832720379572809

Offset: 0

Vaclav Kotesovec, Oct 04 2017

sign

			A293236(n) / ((-1)^n * n!) ~ 1 - 1/n^2 + 3/n^4 + 12/n^5 + 35/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..39

Cf. A293236.

A293071 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} (-1)^jj!x^(j*i).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, 1, 0, 0, 1, -1, 1, 0, 0, 0, 1, -1, 1, -6, 0, 1, 0, 1, -1, 1, -6, 0, -3, 0, 0, 1, -1, 1, -6, 24, 3, 4, 1, 0, 1, -1, 1, -6, 24, 3, 4, -3, 0, 0, 1, -1, 1, -6, 24, -117, -20, -3, 4, 0, 0, 1, -1, 1, -6, 24, -117, -20, -27, -8, -2

Offset: 0

Seiichi Manyama, Oct 03 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0, -1,  1,  1,  1, ...
   0,  0,  0, -6, -6, ...
   0,  0,  0,  0, 24, ...
   0,  1, -3,  3,  3, ...

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

Columns k=0..5 give A000007, A010815, A293072, A293255, A293256, A293257.
Rows n=0 gives A000012.
Main diagonal gives A293236.
Cf. A293202.

nmax = 12;
col[k_] := CoefficientList[Product[Sum[(-1)^j j! x^(i j), {j, 0, k}], {i, 1, nmax+1}] + O[x]^(nmax+1), x]; M = PadRight[col[#], nmax+1]& /@ Range[0, nmax] // Transpose;
A[n_, k_] := M[[n+1, k+1]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 15 2020 *)

A293259 G.f.: Product_{i>0} 1/(Sum_{j>=0} (-1)^jj!x^(j*i)).

Original entry on oeis.org

1, 1, 0, 5, -13, 75, -465, 3509, -29492, 276310, -2854776, 32242512, -395295109, 5230184477, -74303722489, 1128399929626, -18245417102767, 313000130900207, -5678742973964699, 108649510570970878, -2186444702147475131, 46169315317847827548

Offset: 0

Seiichi Manyama, Oct 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A293236, A293251.

nmax = 30; CoefficientList[Series[Product[1/Sum[(-1)^j*j!*x^(j*k), {j, 0, nmax}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 04 2017 *)

Convolution inverse of A293236.

a(n) ~ -(-1)^n * n! * (1 - 2/n - 7/n^3 - 39/n^4 - 272/n^5 - 2457/n^6 - 26443/n^7 - 324675/n^8 - 4453439/n^9 - 67360840/n^10), for coefficients see A293265. - Vaclav Kotesovec, Oct 04 2017

cp's OEIS Frontend

A293267 Coefficients in asymptotic expansion of sequence A293236.

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A293071 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} (-1)^jj!x^(j*i).

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Mathematica

A293259 G.f.: Product_{i>0} 1/(Sum_{j>=0} (-1)^jj!x^(j*i)).

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Mathematica

Formula

cp's OEIS Frontend

A293267 Coefficients in asymptotic expansion of sequence A293236.

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Author

Keywords

Examples

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A293071 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} (-1)^j*j!*x^(j*i).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A293259 G.f.: Product_{i>0} 1/(Sum_{j>=0} (-1)^j*j!*x^(j*i)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A293071 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} Sum_{j=0..k} (-1)^jj!x^(j*i).

A293259 G.f.: Product_{i>0} 1/(Sum_{j>=0} (-1)^jj!x^(j*i)).