OFFSET
1,4
COMMENTS
Also the Bell transform of (-1)^n*(n+1)*Sum_{k=0..n} S1(n,2*k)*(2*k)!/k! where S1 are the Stirling cycle numbers A132393. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 15 2016
LINKS
Indranil Ghosh, Table of n, a(n) for n = Rows 1..50, flattened
FORMULA
T(n,m) = binomial(n,m)*sum(k=0..n-m, ((2*k)!*m^k*stirling1(n-m,2*k))/k!).
EXAMPLE
1,
0, 1,
6, 0, 1,
-24, 24, 0, 1,
170, -120, 60, 0, 1,
-1320, 1380, -360, 120, 0, 1,
11816, -14280, 6090, -840, 210, 0, 1
MATHEMATICA
Flatten[Table[Binomial[n, m]*Sum[((2k)!*m^k*StirlingS1[n-m, 2k])/k!, {k, 0, n-m}], {n, 1, 7}, {m, 1, n}]] (* Indranil Ghosh, Feb 21 2017 *)
PROG
(Maxima)
T(n, m):=binomial(n, m)*sum(((2*k)!*m^k*stirling1(n-m, 2*k))/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 A202185_row(n):
f = lambda n: (-1)^n*(n+1)*sum(factorial(2*k)*stirling_number1(n, 2*k)/ factorial(k) for k in (0..n))
return bell_transform(n, [f(k) for k in (0..n)])
[A202185_row(n) for n in (0..9)] # Peter Luschny, Jan 15 2016
(PARI) T(n, m) = binomial(n, m)*sum(k=0, n-m, ((2*k)!*m^k*stirling(n-m, 2*k, 1))/k!); \\ Michel Marcus, Jan 16 2016
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Vladimir Kruchinin, Dec 13 2011
STATUS
approved