A293133 -id:A293133 - cp's OEIS frontend

A293134 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-x^(k+1)/(1+x)).

1, 1, -1, 1, 0, 3, 1, 0, -2, -13, 1, 0, 0, 6, 73, 1, 0, 0, -6, -12, -501, 1, 0, 0, 0, 24, 0, 4051, 1, 0, 0, 0, -24, -120, 240, -37633, 1, 0, 0, 0, 0, 120, 1080, -2520, 394353, 1, 0, 0, 0, 0, -120, -720, -10080, 21840, -4596553, 1, 0, 0, 0, 0, 0, 720, 5040, 100800

Offset: 0

Seiichi Manyama, Sep 30 2017

			Square array begins:
      1,   1,    1,   1, ...
     -1,   0,    0,   0, ...
      3,  -2,    0,   0, ...
    -13,   6,   -6,   0, ...
     73, -12,   24, -24, ...
   -501,   0, -120, 120, ...

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

Columns k=0..2 give A293125, A293122, A293123.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
A(n,n-1) gives (-1)*A000142(n).
Cf. A293053, A293119, A293133.

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = (-1)^(k+1) * Sum_{i=k..n-1} (-1)^i*(i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.

A293120 Expansion of e.g.f. exp(x^2/(1+x)).

Original entry on oeis.org

1, 0, 2, -6, 36, -240, 1920, -17640, 183120, -2116800, 26943840, -374220000, 5628934080, -91122071040, 1579034096640, -29155689763200, 571308920582400, -11838533804697600, 258608278645516800, -5938673374272038400, 143003892952893772800

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=1 of A293133.
Cf. A111884, A293121.
Cf. A052845.

CoefficientList[Series[E^(x^2/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

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

E.g.f.: exp(x^2/(1+x)).
a(n) = (-1)^n * A052845(n).
a(n) ~ (-1)^n * n^(n-1/4) * exp(-3/2 + 2*sqrt(n) - n)/sqrt(2). - Vaclav Kotesovec, Sep 30 2017

A293121 Expansion of e.g.f. exp(x^3/(1+x)).

Original entry on oeis.org

1, 0, 0, 6, -24, 120, -360, 0, 20160, -302400, 3628800, -39916800, 419126400, -4151347200, 36324288000, -207048441600, -1743565824000, 103742166528000, -2925529096089600, 69945932735078400, -1571249213614080000, 34354603773794304000, -741528257908838400000

Offset: 0

Seiichi Manyama, Sep 30 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..450

Column k=2 of A293133.
Cf. A111884, A293120.
Cf. A293118.

CoefficientList[Series[E^(x^3/(1+x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

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

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

a(n) = (-1)^n * A293118(n).

cp's OEIS Frontend

A293134 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-x^(k+1)/(1+x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A293120 Expansion of e.g.f. exp(x^2/(1+x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A293121 Expansion of e.g.f. exp(x^3/(1+x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula