A351779 -id:A351779

Search: a351779 -id:a351779

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A351776

Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.

+10
5

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,8

LINKS

Table of n, a(n) for n=0..65.

FORMULA

E.g.f. of column k: 1/(1 + k*x*exp(x)).

T(0,k) = 1 and T(n,k) = -k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, ...

0, -1, -2, -3, -4, -5, ...

0, 0, 4, 12, 24, 40, ...

0, 3, -6, -63, -204, -465, ...

0, -4, -8, 420, 2288, 7180, ...

0, -25, 150, -3435, -32020, -138525, ...

PROG

(PARI) T(n, k) = n!*sum(j=0, n, (-k)^(n-j)*(n-j)^j/j!);

(PARI) T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

CROSSREFS

Columns k=0..3 give A000007, A302397, A351777, A351778.

Main diagonal gives A351779.

Cf. A292861, A351761.

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Feb 19 2022

STATUS

approved

A351765

a(n) = n! * Sum_{k=0..n} n^(n-k) * (n-k)^k/k!.

+10
4

1, 1, 12, 279, 11536, 746525, 69768036, 8902181575, 1487939919936, 315597946293657, 82839437215344100, 26366747854082944451, 10006618140321691249296, 4464690010732922712332149, 2313871692128866349730705924, 1378552938661073773617331110975

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..15.

FORMULA

a(n) = n! * [x^n] 1/(1 - n*x*exp(x)).

From Vaclav Kotesovec, Feb 19 2022: (Start)

a(n) ~ exp(1) * n! * n^n.

a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n-1). (End)

MATHEMATICA

Join[{1}, Table[n!*Sum[n^(n - k)*(n - k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 19 2022 *)

PROG

(PARI) a(n) = n!*sum(k=0, n, n^(n-k)*(n-k)^k/k!);

CROSSREFS

Main diagonal of A351761.

Cf. A332408, A351768, A351779.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Feb 18 2022

STATUS

approved

A351780

a(n) = n! * Sum_{k=0..n} (-k)^(n-k) * (n-k)^k/k!.

+10
3

1, 0, -2, 6, 108, -2420, 8730, 1313718, -57930152, 567983736, 109544982390, -9917916180590, 321821829728388, 32383946348733252, -6591798188344856942, 531702135210365508270, 11136706526396459006640

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..16.

PROG

(PARI) a(n) = n!*sum(k=0, n, (-k)^(n-k)*(n-k)^k/k!);

CROSSREFS

Cf. A351768, A351779.

KEYWORD

sign

AUTHOR

Seiichi Manyama, Feb 19 2022

STATUS

approved

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