A336959 -id:A336959

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A336958

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

+10
3

1, -1, -2, 10, 24, -312, -560, 19472, 6272, -1994624, 4072704, 299059968, -1635814400, -60723321856, 628215191552, 15716076562432, -274420622327808, -4900668238036992, 140182198527655936, 1717697481518809088, -83651335147070685184, -590374211868638314496

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

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..21.

FORMULA

a(n) = n! * Sum_{k=0..n} (-2 * (n-k+1))^k / k!.

a(0) = 1; a(n) = -n * a(n-1) - Sum_{k=2..n} binomial(n,k) * 2^k * a(n-k).

MATHEMATICA

nmax = 21; CoefficientList[Series[1/(Exp[2 x] - x), {x, 0, nmax}], x] Range[0, nmax]!

Table[n! Sum[(-2 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 21}]

a[0] = 1; a[n_] := a[n] = -n a[n - 1] - Sum[Binomial[n, k] 2^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 21}]

CROSSREFS

Cf. A089148, A336947, A336959.

KEYWORD

sign

AUTHOR

Ilya Gutkovskiy, Aug 09 2020

STATUS

approved

A351791

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))^j/j!.

+10
3

1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -2, -3, 24, 1, 1, -4, -6, -4, 120, 1, 1, -6, -3, 40, 25, 720, 1, 1, -8, 6, 132, 120, 114, 5040, 1, 1, -10, 21, 248, -375, -1872, -287, 40320, 1, 1, -12, 42, 364, -2120, -8298, -3920, -4152, 362880, 1, 1, -14, 69, 456, -5655, -12144, 86121, 155776, -1647, 3628800

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

OFFSET

0,6

LINKS

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

FORMULA

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

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

EXAMPLE

Square array begins:

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

2, 0, -2, -4, -6, -8, ...

6, -3, -6, -3, 6, 21, ...

24, -4, 40, 132, 248, 364, ...

120, 25, 120, -375, -2120, -5655, ...

MATHEMATICA

T[n_, k_] := n!*(1 + Sum[(-k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)

PROG

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

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

CROSSREFS

Columns k=0..4 give A000142, (-1)^n * A302397(n), A336959, A351792, A351793.

Main diagonal gives (-1)^n * A302398(n).

Cf. A351776, A351790.

KEYWORD

sign,tabl

AUTHOR

Seiichi Manyama, Feb 19 2022

STATUS

approved

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