The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 27 08:50 EDT 2020. Contains 337380 sequences. (Running on oeis4.)