A293015 -id:A293015 - cp's OEIS frontend

A292952 E.g.f.: exp(-x * exp(x)).

1, -1, -1, 2, 9, 4, -95, -414, 49, 10088, 55521, 13870, -2024759, -15787188, -28612415, 616876274, 7476967905, 32522642896, -209513308607, -4924388011050, -38993940088199, -11731860520780, 3807154270837281, 52018152493218010, 278413297030360273

Offset: 0

Seiichi Manyama, Sep 27 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..557

Column k=1 of A293015.
Cf. this sequence (k=1), A292953 (k=2), A292954 (k=3), A292955 (k=4).
Cf. A003725.

With[{nn=30},CoefficientList[Series[Exp[-x Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 15 2023 *)

x='x+O('x^66); Vec(serlaplace(exp(-x*exp(x))))

a(n) = (-1)^n * A003725(n).

A292953 E.g.f.: exp(-1/2! * x^2 * exp(x)).

Original entry on oeis.org

1, 0, -1, -3, -3, 20, 150, 504, -343, -18180, -140220, -500500, 2032899, 50210082, 441768236, 1740141480, -13025325615, -330558552376, -3452606080848, -16648495695792, 136964192085395, 4315989335784630, 55121200672923924, 352945156766431592

Offset: 0

Seiichi Manyama, Sep 27 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..541

Column k=2 of A293015.

Cf. A292952 (k=1), this sequence (k=2), A292954 (k=3), A292955 (k=4).

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

A292954 E.g.f.: exp(-1/3! * x^3 * exp(x)).

Original entry on oeis.org

1, 0, 0, -1, -4, -10, -10, 105, 1064, 6356, 25080, 9075, -1056660, -13219206, -106106364, -548948855, 139658960, 48411569800, 761039099824, 7815284148711, 52216924707660, 9385130453790, -6650556642220260, -132749143322588331, -1713641693856894824

Offset: 0

Seiichi Manyama, Sep 27 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..536

Column k=3 of A293015.
Cf. A292952 (k=1), A292953 (k=2), this sequence (k=3), A292955 (k=4).
Cf. A292910.

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

a(n) = (-1)^n * A292910(n).

A292955 E.g.f.: exp(-1/4! * x^4 * exp(x)).

Original entry on oeis.org

1, 0, 0, 0, -1, -5, -15, -35, -35, 504, 6090, 45870, 270930, 1215500, 1995994, -42118440, -733409495, -8069463780, -70153266240, -468024155016, -1498366231020, 21132982355940, 568009017066260, 8607952077741940, 101448276642079059, 937291639168833850

Offset: 0

Seiichi Manyama, Sep 27 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..538

Column k=4 of A293015.

Cf. A292952 (k=1), A292953 (k=2), A292954 (k=3), this sequence (k=4).

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

A293019 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = - k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, 0, -2, 2, 1, 1, 0, 0, -6, 9, -2, 1, 0, 0, -6, 0, 4, -9, 1, 0, 0, 0, -24, 100, -95, -9, 1, 0, 0, 0, -24, -60, 570, -414, 50, 1, 0, 0, 0, 0, -120, 240, 798, 49, 267, 1, 0, 0, 0, 0, -120, -360, 4830, -15176, 10088, 413, 1, 0, 0

Offset: 0

Seiichi Manyama, Sep 28 2017

			Square array begins:
    1,  1,  1,   1,   1, ...
   -1, -1,  0,   0,   0, ...
    0, -1, -2,   0,   0, ...
    1,  2, -6,  -6,   0, ...
    1,  9,  0, -24, -24, ...

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

Columns k=0-4 give: A000587, A292952, A293016, A293017, A293018.
Rows n=0 gives A000012.
Cf. A292978, A293015.

cp's OEIS Frontend

A292952 E.g.f.: exp(-x * exp(x)).

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A292953 E.g.f.: exp(-1/2! * x^2 * exp(x)).

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Programs

PARI

A292954 E.g.f.: exp(-1/3! * x^3 * exp(x)).

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Programs

PARI

Formula

A292955 E.g.f.: exp(-1/4! * x^4 * exp(x)).

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PARI

A293019 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = - k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

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