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A258773 Triangle read by rows, T(n,k) = (-1)^(n-k)*C(n,k)*k^n, for n>=0 and 0<=k<=n. 2
1, 0, 1, 0, -2, 4, 0, 3, -24, 27, 0, -4, 96, -324, 256, 0, 5, -320, 2430, -5120, 3125, 0, -6, 960, -14580, 61440, -93750, 46656, 0, 7, -2688, 76545, -573440, 1640625, -1959552, 823543, 0, -8, 7168, -367416, 4587520, -21875000, 47029248, -46118408, 16777216 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1326

FORMULA

Sum_{k=0..n} T(n,k) = n!.

Sum_{k=0..n} |T(n,k)| = A072034(n).

Sum_{n>=0} Sum_{k=0..n} T(n,k) x^k y^n/n! = 1/(1 + W(-x*y*exp(-y)) where W is the Lambert W function. - Robert Israel, Dec 16 2015

T(n,n) = A000312(n). - Peter Luschny, Dec 17 2015

T(n, k+1) = n * A075513(n, k) if n>0. - Michael Somos, May 13 2018

EXAMPLE

[1]

[0,  1]

[0, -2,     4]

[0,  3,   -24,     27]

[0, -4,    96,   -324,     256]

[0,  5,  -320,   2430,   -5120,    3125]

[0, -6,   960, -14580,   61440,  -93750,    46656]

[0,  7, -2688,  76545, -573440, 1640625, -1959552, 823543]

MAPLE

seq(seq((-1)^(n-k)*binomial(n, k)*k^n, k=0..n), n=0..8);

T_row := proc(n) (-1)^n*(1-exp(x))^n/n!; diff(%, [x$n]); subs(exp(x)=t, n!*expand(%, x)); CoefficientList(%, t) end: seq(print(T_row(n)), n=0..7);

MATHEMATICA

Flatten@Table[(-1)^(n - k) Binomial[n, k] k^n, {n, 0 , 10}, {k, 0, n}] (* G. C. Greubel, Dec 17 2015 *)

CROSSREFS

Cf. A000142, A000312, A072034, A075513..

Sequence in context: A198543 A294846 A011994 * A054003 A134352 A152648

Adjacent sequences:  A258770 A258771 A258772 * A258774 A258775 A258776

KEYWORD

sign,tabl

AUTHOR

Peter Luschny, Jun 09 2015

STATUS

approved

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Last modified April 5 23:05 EDT 2020. Contains 333260 sequences. (Running on oeis4.)