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A176663 T(n, k) = [x^k] Sum_{j=0..n} j!*binomial(x, j), for 0 <= k <= n, triangle read by rows. 1
1, 1, 1, 1, 0, 1, 1, 2, -2, 1, 1, -4, 9, -5, 1, 1, 20, -41, 30, -9, 1, 1, -100, 233, -195, 76, -14, 1, 1, 620, -1531, 1429, -659, 161, -20, 1, 1, -4420, 11537, -11703, 6110, -1799, 302, -27, 1, 1, 35900, -98047, 106421, -61174, 20650, -4234, 519, -35, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

LINKS

Table of n, a(n) for n=0..54.

FORMULA

From Peter Luschny, Jul 02 2019: (Start)

Sum_{k=0..n) T(n, k) x^k = Sum_{k=0..n) (x)_k, where (x)_k denotes the falling factorial.

Let T be the lower triangular matrix associated to the T(n, k) and S the lower triangular matrix associated to the Stirling set numbers S2(n, k). Then S*T = A186020 (seen as a matrix) and T*S = A000012 (seen as a matrix). (End)

EXAMPLE

Triangle starts:

{1},

{1, 1},

{1, 0, 1},

{1, 2, -2, 1},

{1, -4, 9, -5, 1},

{1, 20, -41, 30, -9, 1},

{1, -100, 233, -195, 76, -14, 1},

{1, 620, -1531, 1429, -659, 161, -20, 1},

{1, -4420, 11537, -11703, 6110, -1799, 302, -27, 1},

{1, 35900, -98047, 106421, -61174, 20650, -4234, 519, -35, 1},

{1, -326980, 928529, -1066279, 662506, -248675, 59039, -8931, 835, -44, 1}

MAPLE

with(PolynomialTools):

T_row := n -> CoefficientList(expand(add(k!*binomial(x, k), k=0..n)), x):

ListTools:-Flatten([seq(T_row(n), n=0..9)]); # Peter Luschny, Jul 02 2019

MATHEMATICA

p[x_, n_] := Sum[k! Binomial[x, k], {k, 0, n}];

Table[CoefficientList[FunctionExpand[p[x, n]], x], {n, 0, 10}] // Flatten

(* Alternative: *)

Table[CoefficientList[FunctionExpand[Sum[FactorialPower[x, k], {k, 0, n}]], x], {n, 0, 10}] // Flatten (* Peter Luschny, Jul 02 2019 *)

CROSSREFS

Row sums are A040000. Alternating row sums are A058006, which are also T(n,1).

Cf. 186020.

Sequence in context: A121697 A225201 A124976 * A113021 A298261 A152937

Adjacent sequences:  A176660 A176661 A176662 * A176664 A176665 A176666

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Apr 23 2010

EXTENSIONS

Edited by Peter Luschny, Jul 02 2019

STATUS

approved

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Last modified October 14 14:45 EDT 2019. Contains 328019 sequences. (Running on oeis4.)