A202183 Triangle T(n,m) = coefficient of x^n in expansion of x^m*(x+1)^(m*x) = sum(n>=m, T(n,m) x^n*m!/n!).

1, 0, 1, 6, 0, 1, -12, 24, 0, 1, 100, -60, 60, 0, 1, -540, 960, -180, 120, 0, 1, 4158, -6300, 4620, -420, 210, 0, 1, -33600, 71904, -35280, 15680, -840, 336, 0, 1, 310896, -725760, 557928, -136080, 42840, -1512, 504, 0, 1, -3160080, 8723520, -6652800

Offset

1,4

Comments

Also the Bell transform of (-1)^n*(n+1)!*Sum_{k=0..n} S1(n-k,k)/(n-k)! where S1 are the Stirling cycle numbers A132393. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 15 2016

Links

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)

Formula

T(n,m) = n!/m!*sum(k=0..n-m, (m^k*stirling1(n-m-k,k))/(n-m-k)!).

Example

1,
0, 1,
6, 0, 1,
-12, 24, 0, 1,
100, -60, 60, 0, 1,
-540, 960, -180, 120, 0, 1,
4158, -6300, 4620, -420, 210, 0, 1

Maple

N:= 10: # for rows 1 to N
for m from 1 to N do
   S[m]:= series(x^m*(x+1)^(m*x), x, N+1);
od:
seq(seq(coeff(S[m], x, n)*n!/m!, m=1..n), n=1..N); # Robert Israel, Jan 15 2016

Prog

(Maxima)
T(n, m):=n!/m!*sum((m^k*stirling1(n-m-k, k))/(n-m-k)!, k, 0, n-m);
(Sage) # uses[bell_transform from A264428]
# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
def A202183_row(n):
    f = lambda n: (-1)^n*factorial(n+1)*sum(stirling_number1(n-k, k)/factorial(n-k) for k in (0..n))
    return bell_transform(n, [f(k) for k in (0..n)])
[A202183_row(n) for n in (0..9)] # Peter Luschny, Jan 15 2016
(PARI) T(n, m) = n!/m!*sum(k=0, n-m, (m^k*stirling(n-m-k, k, 1))/(n-m-k)!); \\ Michel Marcus, Jan 16 2016

Crossrefs

Sequence in context: A317446 A137943 A202189 * A227612 A221273 A352607

Adjacent sequences: A202180 A202181 A202182 * A202184 A202185 A202186

Keyword

sign,tabl

Author

Vladimir Kruchinin, Dec 13 2011

Status

approved